Module: sage.rings.number_field.number_field
Number Fields
Author Log:
NOTE:
Unlike in PARI/GP, class group computations *in SAGE* do *not* by
default assume the Generalized Riemann Hypothesis. To do class
groups computations not provably correctly you must often pass the
flag proof=False to functions or call the function
proof.number_field(False)
. It can easily take 1000's of
times longer to do computations with proof=True
(the
default).
This example follows one in the Magma reference manual:
sage: K.<y> = NumberField(x^4 - 420*x^2 + 40000) sage: z = y^5/11; z 420/11*y^3 - 40000/11*y sage: R.<y> = PolynomialRing(K) sage: f = y^2 + y + 1 sage: L.<a> = K.extension(f); L Number Field in a with defining polynomial y^2 + y + 1 over its base field sage: KL.<b> = NumberField([x^4 - 420*x^2 + 40000, x^2 + x + 1]); KL Number Field in b0 with defining polynomial x^4 + (-420)*x^2 + 40000 over its base field
We do some arithmetic in a tower of relative number fields:
sage: K.<cuberoot2> = NumberField(x^3 - 2) sage: L.<cuberoot3> = K.extension(x^3 - 3) sage: S.<sqrt2> = L.extension(x^2 - 2) sage: S Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field sage: sqrt2 * cuberoot3 cuberoot3*sqrt2 sage: (sqrt2 + cuberoot3)^5 (20*cuberoot3^2 + 15*cuberoot3 + 4)*sqrt2 + 3*cuberoot3^2 + 20*cuberoot3 + 60 sage: cuberoot2 + cuberoot3 cuberoot3 + cuberoot2 sage: cuberoot2 + cuberoot3 + sqrt2 sqrt2 + cuberoot3 + cuberoot2 sage: (cuberoot2 + cuberoot3 + sqrt2)^2 (2*cuberoot3 + 2*cuberoot2)*sqrt2 + cuberoot3^2 + 2*cuberoot2*cuberoot3 + cuberoot2^2 + 2 sage: cuberoot2 + sqrt2 sqrt2 + cuberoot2 sage: a = S(cuberoot2); a cuberoot2 sage: a.parent() Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field
WARNING: Doing arithmetic in towers of relative fields that depends on canonical coercions is currently VERY SLOW. It is much better to explicitly coerce all elements into a common field, then do arithmetic with them there (which is quite fast).
TESTS:
sage: y = polygen(QQ,'y'); K.<beta> = NumberField([y^3 - 3, y^2 - 2]) sage: K(y^10) (-3024*beta1 + 1530)*beta0^2 + (-2320*beta1 + 5067)*beta0 - 3150*beta1 + 7592
Module-level Functions
n, [names=None]) |
Return the n-th cyclotomic field, where n is a positive integer.
Input:
We create the
th cyclotomic field
with the
default generator name.
sage: k = CyclotomicField(7); k Cyclotomic Field of order 7 and degree 6 sage: k.gen() zeta7
Cyclotomic fields are of a special type.
sage: type(k) <class 'sage.rings.number_field.number_field.NumberField_cyclotomic'>
We can specify a different generator name as follows.
sage: k.<z7> = CyclotomicField(7); k Cyclotomic Field of order 7 and degree 6 sage: k.gen() z7
The
must be an integer.
sage: CyclotomicField(3/2) Traceback (most recent call last): ... TypeError: no coercion of this rational to integer
The degree must be positive.
sage: CyclotomicField(0) Traceback (most recent call last): ... ValueError: n (=0) must be a positive integer
The special case
does not return the rational numbers:
sage: CyclotomicField(1) Cyclotomic Field of order 1 and degree 1
sage: cf6 = CyclotomicField(6) ; z6 = cf6.0 sage: cf3 = CyclotomicField(3) ; z3 = cf3.0 sage: cf3(z6) zeta3 + 1 sage: cf6(z3) zeta6 - 1 sage: cf9 = CyclotomicField(9) ; z9 = cf9.0 sage: cf18 = CyclotomicField(18) ; z18 = cf18.0 sage: cf18(z9) zeta18^2 sage: cf9(z18) -zeta9^5 sage: cf18(z3) zeta18^3 - 1 sage: cf18(z6) zeta18^3 sage: cf18(z6)**2 zeta18^3 - 1 sage: cf9(z3) zeta9^3
polynomial, [name=None], [check=True], [names=None], [cache=True]) |
Return the number field defined by the given irreducible polynomial and with variable with the given name. If check is True (the default), also verify that the defining polynomial is irreducible and over Q.
Input:
sage: z = QQ['z'].0 sage: K = NumberField(z^2 - 2,'s'); K Number Field in s with defining polynomial z^2 - 2 sage: s = K.0; s s sage: s*s 2 sage: s^2 2
Constructing a relative number field
sage: K.<a> = NumberField(x^2 - 2) sage: R.<t> = K[] sage: L.<b> = K.extension(t^3+t+a); L Number Field in b with defining polynomial t^3 + t + a over its base field sage: L.absolute_field('c') Number Field in c with defining polynomial x^6 + 2*x^4 + x^2 - 2 sage: a*b a*b sage: L(a) a sage: L.lift_to_base(b^3 + b) -a
Constructing another number field:
sage: k.<i> = NumberField(x^2 + 1) sage: R.<z> = k[] sage: m.<j> = NumberField(z^3 + i*z + 3) sage: m Number Field in j with defining polynomial z^3 + i*z + 3 over its base field
Number fields are globally unique.
sage: K.<a>= NumberField(x^3-5) sage: a^3 5 sage: L.<a>= NumberField(x^3-5) sage: K is L True
Having different defining polynomials makes them fields different:
sage: x = polygen(QQ, 'x'); y = polygen(QQ, 'y') sage: k.<a> = NumberField(x^2 + 3) sage: m.<a> = NumberField(y^2 + 3) sage: k Number Field in a with defining polynomial x^2 + 3 sage: m Number Field in a with defining polynomial y^2 + 3
An example involving a variable name that defines a function in PARI:
sage: theta = polygen(QQ, 'theta') sage: M.<z> = NumberField([theta^3 + 4, theta^2 + 3]); M Number Field in z0 with defining polynomial theta^3 + 4 over its base field
v, names, [check=True]) |
Return the tower of number fields defined by the polynomials or number fields in the list v.
This is the field constructed first from v[0], then over that field from v[1], etc. If all is False, then each v[i] must be irreducible over the previous fields. Otherwise a list of all possible fields defined by all those polynomials is output.
If names defines a variable name a, say, then the generators of the intermediate number fields are a0, a1, a2, ...
Input:
sage: k.<a,b,c> = NumberField([x^2 + 1, x^2 + 3, x^2 + 5]); k Number Field in a with defining polynomial x^2 + 1 over its base field sage: a^2 -1 sage: b^2 -3 sage: c^2 -5 sage: (a+b+c)^2 (2*b + 2*c)*a + 2*c*b - 9
The Galois group is a product of 3 groups of order 2:
sage: k.galois_group() Galois group PARI group [8, 1, 3, "E(8)=2[x]2[x]2"] of degree 8 of the Number Field in a with defining polynomial x^2 + 1 over its base field
Repeatedly calling base_field allows us to descend the internally constructed tower of fields:
sage: k.base_field() Number Field in b with defining polynomial x^2 + 3 over its base field sage: k.base_field().base_field() Number Field in c with defining polynomial x^2 + 5 sage: k.base_field().base_field().base_field() Rational Field
In the following example the second polynomial is reducible over the first, so we get an error:
sage: v = NumberField([x^3 - 2, x^3 - 2], names='a') Traceback (most recent call last): ... ValueError: defining polynomial (x^3 - 2) must be irreducible
We mix polynomial parent rings:
sage: k.<y> = QQ[] sage: m = NumberField([y^3 - 3, x^2 + x + 1, y^3 + 2], 'beta') sage: m Number Field in beta0 with defining polynomial y^3 - 3 over its base field sage: m.base_field () Number Field in beta1 with defining polynomial x^2 + x + 1 over its base field
A tower of quadratic fields:
sage: K.<a> = NumberField([x^2 + 3, x^2 + 2, x^2 + 1]) sage: K Number Field in a0 with defining polynomial x^2 + 3 over its base field sage: K.base_field() Number Field in a1 with defining polynomial x^2 + 2 over its base field sage: K.base_field().base_field() Number Field in a2 with defining polynomial x^2 + 1
A bigger tower of quadratic fields.
sage: K.<a2,a3,a5,a7> = NumberField([x^2 + p for p in [2,3,5,7]]); K Number Field in a2 with defining polynomial x^2 + 2 over its base field sage: a2^2 -2 sage: a3^2 -3 sage: (a2+a3+a5+a7)^3 ((6*a5 + 6*a7)*a3 + 6*a7*a5 - 47)*a2 + (6*a7*a5 - 45)*a3 + (-41)*a5 - 37*a7
poly, name, latex_name) |
This is used in pickling generic number fields.
sage: from sage.rings.number_field.number_field import NumberField_generic_v1 sage: R.<x> = QQ[] sage: NumberField_generic_v1(x^2 + 1, 'i', 'i') Number Field in i with defining polynomial x^2 + 1
zeta_order, name) |
This is used in pickling cyclotomic fields.
sage: from sage.rings.number_field.number_field import NumberField_cyclotomic_v1 sage: NumberField_cyclotomic_v1(5,'a') Cyclotomic Field of order 5 and degree 4 sage: NumberField_cyclotomic_v1(5,'a').variable_name() 'a'
base_field, poly, name, latex_name) |
This is used in pickling relative fields.
sage: from sage.rings.number_field.number_field import NumberField_relative_v1 sage: R.<x> = CyclotomicField(3)[] sage: NumberField_relative_v1(CyclotomicField(3), x^2 + 7, 'a', 'a') Number Field in a with defining polynomial x^2 + 7 over its base field
poly, name, latex_name) |
This is used in pickling generic number fields.
sage: from sage.rings.number_field.number_field import NumberField_generic_v1 sage: R.<x> = QQ[] sage: NumberField_generic_v1(x^2 + 1, 'i', 'i') Number Field in i with defining polynomial x^2 + 1
poly, name) |
This is used in pickling quadratic fields.
sage: from sage.rings.number_field.number_field import NumberField_quadratic_v1 sage: R.<x> = QQ[] sage: NumberField_quadratic_v1(x^2 - 2, 'd') Number Field in d with defining polynomial x^2 - 2
base_field, poly, name, latex_name) |
This is used in pickling relative fields.
sage: from sage.rings.number_field.number_field import NumberField_relative_v1 sage: R.<x> = CyclotomicField(3)[] sage: NumberField_relative_v1(CyclotomicField(3), x^2 + 7, 'a', 'a') Number Field in a with defining polynomial x^2 + 7 over its base field
D, names, [check=True]) |
Return a quadratic field obtained by adjoining a square root of
to the rational numbers, where
is not a perfect square.
Input:
sage: QuadraticField(3, 'a') Number Field in a with defining polynomial x^2 - 3 sage: K.<theta> = QuadraticField(3); K Number Field in theta with defining polynomial x^2 - 3 sage: QuadraticField(9, 'a') Traceback (most recent call last): ... ValueError: D must not be a perfect square. sage: QuadraticField(9, 'a', check=False) Number Field in a with defining polynomial x^2 - 9
Quadratic number fields derive from general number fields.
sage: type(K) <class 'sage.rings.number_field.number_field.NumberField_quadratic'> sage: is_NumberField(K) True
) |
Return the unique copy of the gp (PARI) interpreter used for number field computations.
sage: from sage.rings.number_field.number_field import gp sage: gp() GP/PARI interpreter
x) |
Return True if x is an absolute number field.
sage: is_AbsoluteNumberField(NumberField(x^2+1,'a')) True sage: is_AbsoluteNumberField(NumberField([x^3 + 17, x^2+1],'a')) False
The rationals are a number field, but they're not of the absolute number field class.
sage: is_AbsoluteNumberField(QQ) False
x) |
Return True if x is a cyclotomic field, i.e., of the special cyclotomic field class. This function does not return True for a number field that just happens to be isomorphic to a cyclotomic field.
sage: is_CyclotomicField(NumberField(x^2 + 1,'zeta4')) False sage: is_CyclotomicField(CyclotomicField(4)) True sage: is_CyclotomicField(CyclotomicField(1)) True sage: is_CyclotomicField(QQ) False sage: is_CyclotomicField(7) False
x) |
Return True if x is of the quadratic number field type.
sage: is_QuadraticField(QuadraticField(5,'a')) True sage: is_QuadraticField(NumberField(x^2 - 5, 'b')) True sage: is_QuadraticField(NumberField(x^3 - 5, 'b')) False
A quadratic field specially refers to a number field, not a finite field:
sage: is_QuadraticField(GF(9,'a')) False
x) |
Return True if x is a relative number field.
sage: is_RelativeNumberField(NumberField(x^2+1,'a')) False sage: k.<a> = NumberField(x^3 - 2) sage: l.<b> = k.extension(x^3 - 3); l Number Field in b with defining polynomial x^3 - 3 over its base field sage: is_RelativeNumberField(l) True sage: is_RelativeNumberField(QQ) False
D) |
Return True if the integer
is a fundamental discriminant, i.e.,
if
, and
and either (1)
is square free
or (2) we have
with
and
square free. These are exactly the discriminants of quadratic fields.
sage: [D for D in range(-15,15) if is_fundamental_discriminant(D)] [-15, -11, -8, -7, -4, -3, 5, 8, 12, 13] sage: [D for D in range(-15,15) if not is_square(D) and QuadraticField(D,'a').disc() == D] [-15, -11, -8, -7, -4, -3, 5, 8, 12, 13]
t) |
Used for easily determining the correct proof flag to use.
v) |
Class: NumberField_absolute
self, polynomial, name, [latex_name=None], [check=True]) |
Functions: absolute_polynomial,
absolute_vector_space,
base_field,
change_names,
embeddings,
galois_closure,
is_absolute,
maximal_order,
Minkowski_embedding,
optimized_representation,
optimized_subfields,
order,
places,
real_places,
relativize,
subfields,
vector_space
self) |
Return absolute polynomial that defines this absolute field.
This is the same as self.polynomial()
.
sage: K.<a> = NumberField(x^2 + 1) sage: K.absolute_polynomial () x^2 + 1
self) |
Return vector space over
corresponding to this number
field, along with maps from that space to this number field
and in the other direction.
For an absolute extension this is identical to
self.vector_space()
.
sage: K.<a> = NumberField(x^3 - 5) sage: K.absolute_vector_space() (Vector space of dimension 3 over Rational Field, Isomorphism from Vector space of dimension 3 over Rational Field to Number Field in a with defining polynomial x^3 - 5, Isomorphism from Number Field in a with defining polynomial x^3 - 5 to Vector space of dimension 3 over Rational Field)
self) |
Returns the base field of self, which is always QQ
sage: K = CyclotomicField(5) sage: K.base_field() Rational Field
self, names) |
Return number field isomorphic to self but with the given generator name.
Input:
Also, K.structure()
returns from_K and to_K, where
from_K is an isomorphism from K to self and to_K is an
isomorphism from self to K.
sage: K.<z> = NumberField(x^2 + 3); K Number Field in z with defining polynomial x^2 + 3 sage: L.<ww> = K.change_names() sage: L Number Field in ww with defining polynomial x^2 + 3 sage: L.structure()[0] Number field isomorphism from Number Field in ww with defining polynomial x^2 + 3 to Number Field in z with defining polynomial x^2 + 3 given by variable name change sage: L.structure()[0](ww + 5/3) z + 5/3
self, K) |
Compute all field embeddings of self into the field K (which need not even be a number field, e.g., it could be the complex numbers). This will return an identical result when given K as input again.
If possible, the most natural embedding of K into self is put first in the list.
Input:
sage: K.<a> = NumberField(x^3 - 2) sage: L.<a1> = K.galois_closure(); L Number Field in a1 with defining polynomial x^6 + 40*x^3 + 1372 sage: K.embeddings(L)[0] Ring morphism: From: Number Field in a with defining polynomial x^3 - 2 To: Number Field in a1 with defining polynomial x^6 + 40*x^3 + 1372 Defn: a |--> 1/84*a1^4 + 13/42*a1 sage: K.embeddings(L) is K.embeddings(L) True
We embed a quadratic field into a cyclotomic field:
sage: L.<a> = QuadraticField(-7) sage: K = CyclotomicField(7) sage: L.embeddings(K) [ Ring morphism: From: Number Field in a with defining polynomial x^2 + 7 To: Cyclotomic Field of order 7 and degree 6 Defn: a |--> 2*zeta7^4 + 2*zeta7^2 + 2*zeta7 + 1, Ring morphism: From: Number Field in a with defining polynomial x^2 + 7 To: Cyclotomic Field of order 7 and degree 6 Defn: a |--> -2*zeta7^4 - 2*zeta7^2 - 2*zeta7 - 1 ]
We embed a cubic field in the complex numbers:
sage: K.<a> = NumberField(x^3 - 2) sage: K.embeddings(CC) [ Ring morphism: From: Number Field in a with defining polynomial x^3 - 2 To: Complex Field with 53 bits of precision Defn: a |--> -0.62996052494743... - 1.09112363597172*I, Ring morphism: From: Number Field in a with defining polynomial x^3 - 2 To: Complex Field with 53 bits of precision Defn: a |--> -0.62996052494743... + 1.09112363597172*I, Ring morphism: From: Number Field in a with defining polynomial x^3 - 2 To: Complex Field with 53 bits of precision Defn: a |--> 1.25992104989487 ]
self, [names=None]) |
Return number field
that is the Galois closure of self,
i.e., is generated by all roots of the defining polynomial of
self
Input:
sage: K.<a> = NumberField(x^4 - 2) sage: M = K.galois_closure('b'); M Number Field in b with defining polynomial x^8 + 28*x^4 + 2500 sage: L.<a2> = K.galois_closure(); L Number Field in a2 with defining polynomial x^8 + 28*x^4 + 2500 sage: K.galois_group().order() 8
sage: phi = K.embeddings(L)[0] sage: phi(K.0) 1/120*a2^5 + 19/60*a2 sage: phi(K.0).minpoly() x^4 - 2
self) |
Returns True since self is an absolute field.
sage: K = CyclotomicField(5) sage: K.is_absolute() True
self, [v=None]) |
Return the maximal order, i.e., the ring of integers, associated to this number field.
Input:
In this example, the maximal order cannot be generated by a single element.
sage: k.<a> = NumberField(x^3 + x^2 - 2*x+8) sage: o = k.maximal_order() sage: o Maximal Order in Number Field in a with defining polynomial x^3 + x^2 - 2*x + 8
We compute
-maximal orders for several
. Note that computing
a
-maximal order is much faster in general than computing
the maximal order:
sage: p = next_prime(10^22); q = next_prime(10^23) sage: K.<a> = NumberField(x^3 - p*q) sage: K.maximal_order([3]).basis() [1/3*a^2 + 1/3*a + 1/3, a, a^2] sage: K.maximal_order([2]).basis() [1, a, a^2] sage: K.maximal_order([p]).basis() [1, a, a^2] sage: K.maximal_order([q]).basis() [1, a, a^2] sage: K.maximal_order([p,3]).basis() [1/3*a^2 + 1/3*a + 1/3, a, a^2]
An example with bigger discriminant:
sage: p = next_prime(10^97); q = next_prime(10^99) sage: K.<a> = NumberField(x^3 - p*q) sage: K.maximal_order(prime_range(10000)).basis() [1, a, a^2]
self, [B=None], [prec=None]) |
Return an nxn matrix over RDF whose columns are the images of
the basis
of self over
(as vector spaces), where here
is the
generator of self over
, i.e. self.gen(0). If B
is not None, return the images of the vectors in B as the
columns instead. If prec is not None, use RealField(prec)
instead of RDF.
This embedding is the so-called "Minkowski embedding" of a
number field in
: given the
embeddings
of self in
, write
for the real embeddings, and
for choices of one of each
pair of complex conjugate embeddings (in our case, we simply
choose the one where the image of
has positive real
part). Here
is the signature of self. Then the
Minkowski embedding is given by:
x |-> (
,
,
,
,
,
,
,
)
Equivalently, this is an embedding of self in
so that the usual norm on
coincides with
on self.
TODO: This could be much improved by implementing homomorphisms over VectorSpaces.
sage: F.<alpha> = NumberField(x^3+2) sage: F.Minkowski_embedding() # random low-order bits [ 1.00000000000000 -1.25992104989487 1.58740105196820] [ 1.41421356237000 0.890898718138390 -1.12246204830692] [0.000000000000000 1.54308184421368 1.94416129723541] sage: F.Minkowski_embedding([1, alpha+2, alpha^2-alpha]) # random low-order bits [ 1.00000000000000 0.740078950105127 2.84732210186307] [ 1.41421356237000 3.71932584287839 -2.01336076644531] [0.000000000000000 1.54308184421368 0.401079453021736] sage: F.Minkowski_embedding() * (alpha + 2).vector().transpose() # random low-order bits [0.740078950105127] [ 3.71932584287839] [ 1.54308184421368]
self, [names=None], [both_maps=True]) |
Return a field isomorphic to self with a better defining polynomial if possible, along with field isomorphisms from the new field to self and from self to the new field.
We construct a compositum of 3 quadratic fields, then find an optimized representation and transform elements back and forth.
sage: K = NumberField([x^2 + p for p in [5, 3, 2]],'a').absolute_field('b'); K Number Field in b with defining polynomial x^8 + 40*x^6 + 352*x^4 + 960*x^2 + 576 sage: L, from_L, to_L = K.optimized_representation() sage: L # your answer may different, since algorithm is random Number Field in a14 with defining polynomial x^8 + 4*x^6 + 7*x^4 + 36*x^2 + 81 sage: to_L(K.0) # random 4/189*a14^7 - 1/63*a14^6 + 1/27*a14^5 + 2/9*a14^4 - 5/27*a14^3 + 8/9*a14^2 + 3/7*a14 + 3/7 sage: from_L(L.0) # random 1/1152*a1^7 + 1/192*a1^6 + 23/576*a1^5 + 17/96*a1^4 + 37/72*a1^3 + 5/6*a1^2 + 55/24*a1 + 3/4
The transformation maps are mutually inverse isomorphisms.
sage: from_L(to_L(K.0)) b sage: to_L(from_L(L.0)) # random a14
self, [degree=0], [name=None], [both_maps=True]) |
Return optimized representations of many (but *not* necessarily all!) subfields of self of degree 0, or of all possible degrees if degree is 0.
sage: K = NumberField([x^2 + p for p in [5, 3, 2]],'a').absolute_field('b'); K Number Field in b with defining polynomial x^8 + 40*x^6 + 352*x^4 + 960*x^2 + 576 sage: L = K.optimized_subfields(name='b') sage: L[0][0] Number Field in b0 with defining polynomial x - 1 sage: L[1][0] Number Field in b1 with defining polynomial x^2 - x + 1 sage: [z[0] for z in L] # random -- since algorithm is random [Number Field in b0 with defining polynomial x - 1, Number Field in b1 with defining polynomial x^2 - x + 1, Number Field in b2 with defining polynomial x^4 - 5*x^2 + 25, Number Field in b3 with defining polynomial x^4 - 2*x^2 + 4, Number Field in b4 with defining polynomial x^8 + 4*x^6 + 7*x^4 + 36*x^2 + 81]
We examine one of the optimized subfields in more detail:
sage: M, from_M, to_M = L[2] sage: M # random Number Field in b2 with defining polynomial x^4 - 5*x^2 + 25 sage: from_M # may be slightly random Ring morphism: From: Number Field in b2 with defining polynomial x^4 - 5*x^2 + 25 To: Number Field in a1 with defining polynomial x^8 + 40*x^6 + 352*x^4 + 960*x^2 + 576 Defn: b2 |--> -5/1152*a1^7 + 1/96*a1^6 - 97/576*a1^5 + 17/48*a1^4 - 95/72*a1^3 + 17/12*a1^2 - 53/24*a1 - 1
The to_M map is None, since there is no map from K to M:
sage: to_M
We apply the from_M map to the generator of M, which gives a rather
large element of
:
sage: from_M(M.0) # random -5/1152*a1^7 + 1/96*a1^6 - 97/576*a1^5 + 17/48*a1^4 - 95/72*a1^3 + 17/12*a1^2 - 53/24*a1 - 1
Nevertheless, that large-ish element lies in a degree 4 subfield:
sage: from_M(M.0).minpoly() # random x^4 - 5*x^2 + 25
self) |
Return the order with given ring generators in the maximal order of this number field.
Input:
sage: k.<i> = NumberField(x^2 + 1) sage: k.order(2*i) Order in Number Field in i with defining polynomial x^2 + 1 sage: k.order(10*i) Order in Number Field in i with defining polynomial x^2 + 1 sage: k.order(3) Traceback (most recent call last): ... ValueError: the rank of the span of gens is wrong sage: k.order(i/2) Traceback (most recent call last): ... ValueError: each generator must be integral
Alternatively, an order can be constructed by adjoining
elements to
:
self, [all_complex=False], [prec=None]) |
Return the collection of all places of self. By default, this returns the set of real places as homomorphisms into RIF first, followed by a choice of one of each pair of complex conjugate homomorphisms into CIF.
On the other hand, if prec is not None, we simply return places into RealField(prec) and ComplexField(prec) (or RDF, CDF if prec=53).
There is an optional flag all_complex, which defaults to False. If all_complex is True, then the real embeddings are returned as embeddings into CIF instead of RIF.
sage: F.<alpha> = NumberField(x^3-100*x+1) ; F.places() [Ring morphism: From: Number Field in alpha with defining polynomial x^3 - 100*x + 1 To: Real Field with 106 bits of precision Defn: alpha |--> -10.00499625499181184573367219280, Ring morphism: From: Number Field in alpha with defining polynomial x^3 - 100*x + 1 To: Real Field with 106 bits of precision Defn: alpha |--> 0.01000001000003000012000055000273, Ring morphism: From: Number Field in alpha with defining polynomial x^3 - 100*x + 1 To: Real Field with 106 bits of precision Defn: alpha |--> 9.994996244991781845613530439509]
sage: F.<alpha> = NumberField(x^3+7) ; F.places() [Ring morphism: From: Number Field in alpha with defining polynomial x^3 + 7 To: Real Field with 106 bits of precision Defn: alpha |--> -1.912931182772389101199116839549, Ring morphism: From: Number Field in alpha with defining polynomial x^3 + 7 To: Complex Field with 53 bits of precision Defn: alpha |--> 0.956465591386195 + 1.65664699997230*I]
sage: F.<alpha> = NumberField(x^3+7) ; F.places(all_complex=True) [Ring morphism: From: Number Field in alpha with defining polynomial x^3 + 7 To: Complex Field with 53 bits of precision Defn: alpha |--> -1.91293118277239, Ring morphism: From: Number Field in alpha with defining polynomial x^3 + 7 To: Complex Field with 53 bits of precision Defn: alpha |--> 0.956465591386195 + 1.65664699997230*I] sage: F.places(prec=10) [Ring morphism: From: Number Field in alpha with defining polynomial x^3 + 7 To: Real Field with 10 bits of precision Defn: alpha |--> -1.9, Ring morphism: From: Number Field in alpha with defining polynomial x^3 + 7 To: Complex Field with 10 bits of precision Defn: alpha |--> 0.96 + 1.7*I]
self, [prec=None]) |
Return all real places of self as homomorphisms into RIF.
sage: F.<alpha> = NumberField(x^4-7) ; F.real_places() [Ring morphism: From: Number Field in alpha with defining polynomial x^4 - 7 To: Real Field with 106 bits of precision Defn: alpha |--> -1.626576561697785743211232345494, Ring morphism: From: Number Field in alpha with defining polynomial x^4 - 7 To: Real Field with 106 bits of precision Defn: alpha |--> 1.626576561697785743211232345494]
self, alpha, names) |
Given an element alpha in self, return a relative number field
isomorphic to self that is relative over the absolute field
, along with isomorphisms from
to self and
from self to K.
Input:
Also, K.structure()
returns from_K and to_K, where
from_K is an isomorphism from K to self and to_K is an isomorphism
from self to K.
sage: K.<a> = NumberField(x^10 - 2) sage: L.<c,d> = K.relativize(a^4 + a^2 + 2); L Number Field in c with defining polynomial x^2 - 1/5*d^4 + 8/5*d^3 - 23/5*d^2 + 7*d - 18/5 over its base field sage: c.absolute_minpoly() x^10 - 2 sage: d.absolute_minpoly() x^5 - 10*x^4 + 40*x^3 - 90*x^2 + 110*x - 58 sage: (a^4 + a^2 + 2).minpoly() x^5 - 10*x^4 + 40*x^3 - 90*x^2 + 110*x - 58 sage: from_L, to_L = L.structure() sage: to_L(a) c sage: to_L(a^4 + a^2 + 2) d sage: from_L(to_L(a^4 + a^2 + 2)) a^4 + a^2 + 2
self, [degree=0], [name=None]) |
sage: K.<a> = NumberField( [x^3 - 2, x^2 + x + 1] ); sage: K = K.absolute_field('b') sage: S = K.subfields() sage: len(S) 6 sage: [k[0].polynomial() for k in S] [x - 3, x^2 - 3*x + 9, x^3 - 3*x^2 + 3*x + 1, x^3 - 3*x^2 + 3*x + 1, x^3 - 3*x^2 + 3*x - 17, x^6 - 3*x^5 + 6*x^4 - 11*x^3 + 12*x^2 + 3*x + 1]
self) |
Return a vector space V and isomorphisms self -> V and V -> self.
Output:
sage: k.<a> = NumberField(x^3 + 2) sage: V, from_V, to_V = k.vector_space() sage: from_V(V([1,2,3])) 3*a^2 + 2*a + 1 sage: to_V(1 + 2*a + 3*a^2) (1, 2, 3) sage: V Vector space of dimension 3 over Rational Field sage: to_V Isomorphism from Number Field in a with defining polynomial x^3 + 2 to Vector space of dimension 3 over Rational Field sage: from_V(to_V(2/3*a - 5/8)) 2/3*a - 5/8 sage: to_V(from_V(V([0,-1/7,0]))) (0, -1/7, 0)
Special Functions: __init__,
__reduce__,
_subfields_helper
self) |
TESTS:
sage: Z = var('Z') sage: K.<w> = NumberField(Z^3 + Z + 1) sage: L = loads(dumps(K)) sage: print L Number Field in w with defining polynomial Z^3 + Z + 1 sage: print L == K True
Class: NumberField_cyclotomic
The command CyclotomicField(n) creates the n-th cyclotomic field, obtained by adjoining an n-th root of unity to the rational field.
sage: CyclotomicField(3) Cyclotomic Field of order 3 and degree 2 sage: CyclotomicField(18) Cyclotomic Field of order 18 and degree 6 sage: z = CyclotomicField(6).gen(); z zeta6 sage: z^3 -1 sage: (1+z)^3 6*zeta6 - 3
sage: K = CyclotomicField(197) sage: loads(K.dumps()) == K True sage: loads((z^2).dumps()) == z^2 True
sage: cf12 = CyclotomicField( 12 ) sage: z12 = cf12.0 sage: cf6 = CyclotomicField( 6 ) sage: z6 = cf6.0 sage: FF = Frac( cf12['x'] ) sage: x = FF.0 sage: print z6*x^3/(z6 + x) zeta12^2*x^3/(x + zeta12^2)
sage: cf6 = CyclotomicField(6) ; z6 = cf6.gen(0) sage: cf3 = CyclotomicField(3) ; z3 = cf3.gen(0) sage: cf3(z6) zeta3 + 1 sage: cf6(z3) zeta6 - 1 sage: type(cf6(z3)) <type 'sage.rings.number_field.number_field_element_quadratic.NumberFieldEl ement_quadratic'> sage: cf1 = CyclotomicField(1) ; z1 = cf1.0 sage: cf3(z1) 1 sage: type(cf3(z1)) <type 'sage.rings.number_field.number_field_element_quadratic.NumberFieldEl ement_quadratic'>
self, n, names) |
A cyclomotic field, i.e., a field obtained by adjoining an n-th root of unity to the rational numbers.
sage: k = CyclotomicField(3) sage: type(k) <class 'sage.rings.number_field.number_field.NumberField_cyclotomic'>
Functions: complex_embedding,
complex_embeddings,
discriminant,
integral_basis,
is_galois,
is_isomorphic,
next_split_prime,
number_of_roots_of_unity,
real_embeddings,
roots_of_unity,
signature,
zeta,
zeta_order
self, [prec=53]) |
Return the embedding of this cyclotomic field into the
approximate complex field with precision prec obtained by
sending the generator
of self to exp(2*pi*i/n), where
is the multiplicative order of
.
If prec is 53 (the default), then the complex double field is used; otherwise the arbitrary precision (but slow) complex field is used.
sage: C = CyclotomicField(4) sage: C.complex_embedding() Ring morphism: From: Cyclotomic Field of order 4 and degree 2 To: Complex Double Field Defn: zeta4 |--> 6.12323399574e-17 + 1.0*I
Note in the example above that the way zeta is computed (using sin and cosine in MPFR) means that only the prec bits of the number after the decimal point are valid.
sage: K = CyclotomicField(3) sage: phi = K.complex_embedding(10) sage: phi(K.0) -0.50 + 0.87*I sage: phi(K.0^3) 1.0 sage: phi(K.0^3 - 1) 0 sage: phi(K.0^3 + 7) 8.0
self, [prec=53]) |
Return all embeddings of this cyclotomic field into the approximate complex field with precision prec.
If prec is 53 (the default), then the complex double field is
used; otherwise the arbitrary precision (but slow) complex
field is used. If you want 53-bit arbitrary precision then
do self.embeddings(ComplexField(53))
.
sage: CyclotomicField(5).complex_embeddings() [ Ring morphism: From: Cyclotomic Field of order 5 and degree 4 To: Complex Double Field Defn: zeta5 |--> 0.309016994375 + 0.951056516295*I, Ring morphism: From: Cyclotomic Field of order 5 and degree 4 To: Complex Double Field Defn: zeta5 |--> -0.809016994375 + 0.587785252292*I, Ring morphism: From: Cyclotomic Field of order 5 and degree 4 To: Complex Double Field Defn: zeta5 |--> -0.809016994375 - 0.587785252292*I, Ring morphism: From: Cyclotomic Field of order 5 and degree 4 To: Complex Double Field Defn: zeta5 |--> 0.309016994375 - 0.951056516295*I ]
self, [v=None]) |
Returns the discriminant of the ring of integers of the cyclotomic field self, or if v is specified, the determinant of the trace pairing on the elements of the list v.
Uses the formula for the discriminant of a prime power cyclotomic field and Hilbert Theorem 88 on the discriminant of composita.
Input:
sage: CyclotomicField(20).discriminant() 4000000 sage: CyclotomicField(18).discriminant() -19683
self, [v=None]) |
Return a list of elements of this number field that are a basis for the full ring of integers.
This field is cyclomotic, so this is a trivial computation, since the power basis on the generator is an integral basis. Thus the v parameter is ignored.
sage: CyclotomicField(5).integral_basis() [1, zeta5, zeta5^2, zeta5^3]
self) |
Return True since all cyclotomic fields are automatically Galois.
sage: CyclotomicField(29).is_galois() True
self, other) |
Return True if the cyclotomic field self is isomorphic as a number field to other.
sage: CyclotomicField(11).is_isomorphic(CyclotomicField(22)) True sage: CyclotomicField(11).is_isomorphic(CyclotomicField(23)) False sage: CyclotomicField(3).is_isomorphic(NumberField(x^2 + x +1, 'a')) True
self, [p=2]) |
Return the next prime integer
that splits completely in
this cyclotomic field (and does not ramify).
sage: K.<z> = CyclotomicField(3) sage: K.next_split_prime(7) 13
self) |
Return number of roots of unity in this cyclotomic field.
sage: K.<a> = CyclotomicField(21) sage: K.number_of_roots_of_unity() 42
self, [prec=53]) |
Return all embeddings of this cyclotomic field into the approximate real field with precision prec.
If prec is 53 (the default), then the real double field is used; otherwise the arbitrary precision (but slow) real field is used.
Mostly, of course, there are no such embeddings.
sage: CyclotomicField(4).real_embeddings() [] sage: CyclotomicField(2).real_embeddings() [ Ring morphism: From: Cyclotomic Field of order 2 and degree 1 To: Real Double Field Defn: -1 |--> -1.0 ]
self) |
Return all the roots of unity in this cyclotomic field, primitive or not.
sage: K.<a> = CyclotomicField(3) sage: zs = K.roots_of_unity(); zs [1, a, -a - 1, -1, -a, a + 1] sage: [ z**K.number_of_roots_of_unity() for z in zs ] [1, 1, 1, 1, 1, 1]
self) |
Return (r1, r2), where r1 and r2 are the number of real embeddings and pairs of complex embeddings of this cyclotomic field, respectively.
Trivial since, apart from QQ, cyclotomic fields are totally complex.
sage: CyclotomicField(5).signature() (0, 2) sage: CyclotomicField(2).signature() (1, 0)
self, [n=None], [all=False]) |
Returns an element of multiplicative order
in this this
cyclotomic field, if there is one. Raises a ValueError if
there is not.
Input:
sage: k = CyclotomicField(7) sage: k.zeta() zeta7 sage: k.zeta().multiplicative_order() 7 sage: k = CyclotomicField(49) sage: k.zeta().multiplicative_order() 49 sage: k.zeta(7).multiplicative_order() 7 sage: k.zeta() zeta49 sage: k.zeta(7) zeta49^7
sage: K.<a> = CyclotomicField(7) sage: K.zeta(14, all=True) [-a^4, -a^5, a^5 + a^4 + a^3 + a^2 + a + 1, -a, -a^2, -a^3] sage: K.<a> = CyclotomicField(10) sage: K.zeta(20, all=True) Traceback (most recent call last): ... ValueError: n (=20) does not divide order of generator
sage: K.<a> = CyclotomicField(5) sage: K.zeta(4) Traceback (most recent call last): ... ValueError: n (=4) does not divide order of generator sage: v = K.zeta(5, all=True); v [a, a^2, a^3, -a^3 - a^2 - a - 1] sage: [b^5 for b in v] [1, 1, 1, 1]
self) |
Return the order of the maximal root of unity contained in this cyclotomic field.
sage: CyclotomicField(1).zeta_order() 2 sage: CyclotomicField(4).zeta_order() 4 sage: CyclotomicField(5).zeta_order() 10 sage: CyclotomicField(5)._n() 5 sage: CyclotomicField(389).zeta_order() 778
Special Functions: __call__,
__init__,
__reduce__,
_coerce_from_gap,
_coerce_from_other_cyclotomic_field,
_coerce_impl,
_Hom_,
_latex_,
_magma_init_,
_multiplicative_order_table,
_n,
_repr_
self, x) |
Create an element of this cyclotomic field from
.
The following example illustrates coercion from the cyclotomic field Q(zeta_42) to the cyclotomic field Q(zeta_6), in a case where such coercion is defined:
sage: k42 = CyclotomicField(42) sage: k6 = CyclotomicField(6) sage: a = k42.gen(0) sage: b = a^7 sage: b zeta42^7 sage: k6(b) zeta6 sage: b^2 zeta42^7 - 1 sage: k6(b^2) zeta6 - 1
Coercion of GAP cyclotomic elements is also supported.
sage: K.<z> = CyclotomicField(7) sage: O = K.maximal_order() sage: K(O.1) z sage: K(O.1^2 + O.1 - 2) z^2 + z - 2
self) |
TESTS:
sage: K.<zeta7> = CyclotomicField(7) sage: L = loads(dumps(K)) sage: print L Cyclotomic Field of order 7 and degree 6 sage: print L == K True
self, x) |
Attempt to coerce a GAP number field element into this cyclotomic field.
sage: k5.<z> = CyclotomicField(5) sage: gap('E(5)^7 + 3') -3*E(5)-2*E(5)^2-3*E(5)^3-3*E(5)^4 sage: w = gap('E(5)^7 + 3') sage: z^7 + 3 z^2 + 3 sage: k5(w) z^2 + 3
self, x, [only_canonical=False]) |
Coerce an element x of a cyclotomic field into self, if at all possible.
Input:
sage: K = CyclotomicField(24) ; L = CyclotomicField(48) sage: L._coerce_from_other_cyclotomic_field(K.0+1) zeta48^2 + 1 sage: K(L.0**2) zeta24
self, x) |
Canonical implicit coercion of x into self.
Elements of other compatible cyclotomic fields coerce in, as do elements of the rings that coerce to all number fields (e.g., integers, rationals).
sage: CyclotomicField(15)._coerce_impl(CyclotomicField(5).0 - 17/3) zeta15^3 - 17/3 sage: K.<a> = CyclotomicField(16) sage: K(CyclotomicField(4).0) a^4
self, codomain, [cat=None]) |
Return homset of homomorphisms from the cyclotomic field self to the number field codomain.
The cat option is currently ignored.
This function is implicitly caled by the Hom method or function.
sage: K.<a> = NumberField(x^2 + 3); K Number Field in a with defining polynomial x^2 + 3 sage: CyclotomicField(3).Hom(K) Set of field embeddings from Cyclotomic Field of order 3 and degree 2 to Number Field in a with defining polynomial x^2 + 3 sage: End(CyclotomicField(21)) Automorphism group of Cyclotomic Field of order 21 and degree 12
self) |
Return the latex representation of this cyclotomic field.
sage: Z = CyclotomicField(4) sage: Z.gen() zeta4 sage: latex(Z) \mathbf{Q}(\zeta_{4})
Latex printing respects the generator name.
sage: k.<a> = CyclotomicField(4) sage: latex(k) \mathbf{Q}[a]/(a^{2} + 1) sage: k Cyclotomic Field of order 4 and degree 2 sage: k.gen() a
self) |
Return a dictionary that maps powers of zeta to their order. This makes computing the orders of the elements of finite order in this field faster.
sage: v = CyclotomicField(6)._multiplicative_order_table() sage: w = v.items(); w.sort(); w [(-1, 2), (1, 1), (-x, 3), (-x + 1, 6), (x - 1, 3), (x, 6)]
self) |
Return the n used to create this cyclotomic field.
sage: CyclotomicField(3).zeta_order() 6 sage: CyclotomicField(3)._n() 3
self) |
Return string representation of this cyclotomic field.
The ``order'' of the cyclotomic field
in the
string output refers to the order of the
, i.e., it
is the integer
. The degree is the degree of the field as
an extension of
.
sage: CyclotomicField(4)._repr_() 'Cyclotomic Field of order 4 and degree 2' sage: CyclotomicField(400)._repr_() 'Cyclotomic Field of order 400 and degree 160'
Class: NumberField_generic
sage: K.<a> = NumberField(x^3 - 2); K Number Field in a with defining polynomial x^3 - 2 sage: loads(K.dumps()) == K True
self, polynomial, name, [latex_name=None], [check=True]) |
Create a number field.
sage: NumberField(x^97 - 19, 'a') Number Field in a with defining polynomial x^97 - 19
If you use check=False, you avoid checking irreducibility of the defining polynomial, which can save time.
sage: K.<a> = NumberField(x^2 - 1, check=False)
It can also be dangerous:
sage: (a-1)*(a+1) 0
Functions: absolute_degree,
absolute_field,
category,
change_generator,
characteristic,
class_group,
class_number,
complex_embeddings,
composite_fields,
defining_polynomial,
degree,
different,
disc,
discriminant,
elements_of_norm,
extension,
factor,
fractional_ideal,
galois_group,
gen,
ideal,
ideals_of_bdd_norm,
integral_basis,
integral_elements_with_trace,
is_absolute,
is_field,
is_galois,
is_isomorphic,
is_relative,
is_totally_imaginary,
is_totally_real,
latex_variable_name,
narrow_class_group,
ngens,
number_of_roots_of_unity,
order,
pari_bnf,
pari_bnf_certify,
pari_nf,
pari_polynomial,
polynomial,
polynomial_ntl,
polynomial_quotient_ring,
polynomial_ring,
prime_above,
primes_above,
primes_of_degree_one_iter,
primes_of_degree_one_list,
primitive_element,
real_embeddings,
reduced_basis,
reduced_gram_matrix,
regulator,
residue_field,
roots_of_unity,
signature,
structure,
subfield,
trace_pairing,
uniformizer,
units,
zeta,
zeta_coefficients,
zeta_function,
zeta_order
self) |
Return the degree of self over
.
sage: NumberField(x^3 + x^2 + 997*x + 1, 'a').absolute_degree() 3 sage: NumberField(x + 1, 'a').absolute_degree() 1 sage: NumberField(x^997 + 17*x + 3, 'a', check=False).absolute_degree() 997
self, names) |
Returns self as an absolute extension over QQ.
Output:
Also, K.structure()
returns from_K and to_K, where
from_K is an isomorphism from K to self and to_K is an isomorphism
from self to K.
sage: K = CyclotomicField(5) sage: K.absolute_field('a') Number Field in a with defining polynomial x^4 + x^3 + x^2 + x + 1
self) |
Return the category of number fields.
sage: NumberField(x^2 + 3, 'a').category() Category of number fields sage: category(NumberField(x^2 + 3, 'a')) Category of number fields
The special types of number fields, e.g., quadratic fields, don't have their own category:
sage: QuadraticField(2,'d').category() Category of number fields
self, alpha, [name=None]) |
Given the number field self, construct another isomorphic
number field
generated by the element alpha of self, along
with isomorphisms from
to self and from self to
.
sage: K.<i> = NumberField(x^2 + 1); K Number Field in i with defining polynomial x^2 + 1 sage: L.<i> = NumberField(x^2 + 1); L Number Field in i with defining polynomial x^2 + 1 sage: K, from_K, to_K = L.change_generator(i/2 + 3) sage: K Number Field in i0 with defining polynomial x^2 - 6*x + 37/4 sage: from_K Ring morphism: From: Number Field in i0 with defining polynomial x^2 - 6*x + 37/4 To: Number Field in i with defining polynomial x^2 + 1 Defn: i0 |--> 1/2*i + 3 sage: to_K Ring morphism: From: Number Field in i with defining polynomial x^2 + 1 To: Number Field in i0 with defining polynomial x^2 - 6*x + 37/4 Defn: i |--> 2*i0 - 6
We compute the image of the generator
of
.
sage: to_K(L.0) 2*i0 - 6
Note that he image is indeed a square root of -1.
sage: to_K(L.0)^2 -1 sage: from_K(to_K(L.0)) i sage: to_K(from_K(K.0)) i0
self) |
Return the characteristic of this number field, which is of course 0.
sage: k.<a> = NumberField(x^99 + 2); k Number Field in a with defining polynomial x^99 + 2 sage: k.characteristic() 0
self, [proof=None], [names=c]) |
Return the class group of the ring of integers of this number field.
Input:
sage: K.<a> = NumberField(x^2 + 23) sage: G = K.class_group(); G Class group of order 3 with structure C3 of Number Field in a with defining polynomial x^2 + 23 sage: G.0 Fractional ideal class (2, 1/2*a - 1/2) sage: G.gens() [Fractional ideal class (2, 1/2*a - 1/2)]
sage: G.number_field() Number Field in a with defining polynomial x^2 + 23 sage: G is K.class_group() True sage: G is K.class_group(proof=False) False sage: G.gens() [Fractional ideal class (2, 1/2*a - 1/2)]
There can be multiple generators:
sage: k.<a> = NumberField(x^2 + 20072) sage: G = k.class_group(); G Class group of order 76 with structure C38 x C2 of Number Field in a with defining polynomial x^2 + 20072 sage: G.gens() [Fractional ideal class (41, a + 10), Fractional ideal class (2, -1/2*a)] sage: G.0 Fractional ideal class (41, a + 10) sage: G.0^20 Fractional ideal class (43, a + 3) sage: G.0^38 Trivial principal fractional ideal class sage: G.1 Fractional ideal class (2, -1/2*a) sage: G.1^2 Trivial principal fractional ideal class
Class groups of Hecke polynomials tend to be very small:
sage: f = ModularForms(97, 2).T(2).charpoly() sage: f.factor() (x - 3) * (x^3 + 4*x^2 + 3*x - 1) * (x^4 - 3*x^3 - x^2 + 6*x - 1) sage: for g,_ in f.factor(): print NumberField(g,'a').class_group().order() ... 1 1 1
self, [proof=None]) |
Return the class number of this number field, as an integer.
Input:
sage: NumberField(x^2 + 23, 'a').class_number() 3 sage: NumberField(x^2 + 163, 'a').class_number() 1 sage: NumberField(x^3 + x^2 + 997*x + 1, 'a').class_number(proof=False) 1539
self, [prec=53]) |
Return all homomorphisms of this number field into the approximate complex field with precision prec.
If prec is 53 (the default), then the complex double field is
used; otherwise the arbitrary precision (but slow) complex
field is used. If you want 53-bit arbitrary precision then
do self.embeddings(ComplexField(53))
.
sage: k.<a> = NumberField(x^5 + x + 17) sage: v = k.complex_embeddings() sage: [phi(k.0^2) for phi in v] # random low order bits [2.97572074038 + 1.73496602657e-16*I, -2.40889943716 + 1.90254105304*I, -2.40889943716 - 1.90254105304*I, 0.921039066973 + 3.07553311885*I, 0.921039066973 - 3.07553311885*I] sage: K.<a> = NumberField(x^3 + 2) sage: K.complex_embeddings() # random low order bits [ Ring morphism: From: Number Field in a with defining polynomial x^3 + 2 To: Complex Double Field Defn: a |--> -1.25992104989 - 3.88578058619e-16*I, Ring morphism: From: Number Field in a with defining polynomial x^3 + 2 To: Complex Double Field Defn: a |--> 0.629960524947 - 1.09112363597*I, Ring morphism: From: Number Field in a with defining polynomial x^3 + 2 To: Complex Double Field Defn: a |--> 0.629960524947 + 1.09112363597*I ]
self, other, [names=None]) |
List of all possible composite number fields formed from self and other.
Input:
sage: K.<a> = NumberField(x^4 - 2) sage: K.composite_fields(K) [Number Field in a0 with defining polynomial x^4 - 162, Number Field in a1 with defining polynomial x^4 - 2, Number Field in a2 with defining polynomial x^8 + 28*x^4 + 2500] sage: k.<a> = NumberField(x^3 + 2) sage: m.<b> = NumberField(x^3 + 2) sage: k.composite_fields(m, 'c') [Number Field in c0 with defining polynomial x^3 - 2, Number Field in c1 with defining polynomial x^6 - 40*x^3 + 1372]
self) |
Return the defining polynomial of this number field.
This is exactly the same as self.polynomal()
.
sage: k5.<z> = CyclotomicField(5) sage: k5.defining_polynomial() x^4 + x^3 + x^2 + x + 1 sage: y = polygen(QQ,'y') sage: k.<a> = NumberField(y^9 - 3*y + 5); k Number Field in a with defining polynomial y^9 - 3*y + 5 sage: k.defining_polynomial() y^9 - 3*y + 5
self) |
Return the degree of this number field.
sage: NumberField(x^3 + x^2 + 997*x + 1, 'a').degree() 3 sage: NumberField(x + 1, 'a').degree() 1 sage: NumberField(x^997 + 17*x + 3, 'a', check=False).degree() 997
self) |
Compute the different fractional ideal of this number field.
The different is the set of all
in
such that the trace
of
is an integer for all
.
sage: k.<a> = NumberField(x^2 + 23) sage: d = k.different() sage: d # random sign in output Fractional ideal (-a) sage: d.norm() 23 sage: k.disc() -23
The different is cached:
sage: d is k.different() True
Another example:
sage: k.<b> = NumberField(x^2 - 123) sage: d = k.different(); d Fractional ideal (2*b) sage: d.norm() 492 sage: k.disc() 492
self, [v=None]) |
Shortcut for self.discriminant.
sage: k.<b> = NumberField(x^2 - 123) sage: k.disc() 492
self, [v=None]) |
Returns the discriminant of the ring of integers of the number field, or if v is specified, the determinant of the trace pairing on the elements of the list v.
Input:
sage: K.<t> = NumberField(x^3 + x^2 - 2*x + 8) sage: K.disc() -503 sage: K.disc([1, t, t^2]) -2012 sage: K.disc([1/7, (1/5)*t, (1/3)*t^2]) -2012/11025 sage: (5*7*3)^2 11025
self, n, [proof=None]) |
Return a list of solutions modulo units of positive norm to
, where a can be any integer in this number field.
Input:
sage: K.<a> = NumberField(x^2+1) sage: K.elements_of_norm(3) [] sage: K.elements_of_norm(50) [7*a - 1, -5*a + 5, a - 7] # 32-bit [7*a - 1, -5*a + 5, -7*a - 1] # 64-bit
self, poly, [name=None], [names=None], [check=True]) |
Return the relative extension of this field by a given polynomial.
sage: K.<a> = NumberField(x^3 - 2) sage: R.<t> = K[] sage: L.<b> = K.extension(t^2 + a); L Number Field in b with defining polynomial t^2 + a over its base field
We create another extension.
sage: k.<a> = NumberField(x^2 + 1); k Number Field in a with defining polynomial x^2 + 1 sage: y = var('y') sage: m.<b> = k.extension(y^2 + 2); m Number Field in b with defining polynomial y^2 + 2 over its base field
Note that b is a root of
:
sage: b.minpoly() x^2 + 2 sage: b.minpoly('z') z^2 + 2
A relative extension of a relative extension.
sage: k.<a> = NumberField([x^2 + 1, x^3 + x + 1]) sage: R.<z> = k[] sage: L.<b> = NumberField(z^3 + 3 + a); L Number Field in b with defining polynomial z^3 + a0 + 3 over its base field
self, n) |
Ideal factorization of the principal ideal of the ring
of integers generated by
.
Here we show how to factor gaussian integers.
First we form a number field defined by
:
sage: K.<I> = NumberField(x^2 + 1); K Number Field in I with defining polynomial x^2 + 1
Here are the factors:
sage: fi, fj = K.factor(13); fi,fj ((Fractional ideal (-3*I - 2), 1), (Fractional ideal (3*I - 2), 1))
Now we extract the reduced form of the generators:
sage: zi = fi[0].gens_reduced()[0]; zi -3*I - 2 sage: zj = fj[0].gens_reduced()[0]; zj 3*I - 2
We recover the integer that was factored in
sage: zi*zj 13
One can also factor elements of the number field:
sage: K.<a> = NumberField(x^2 + 1) sage: K.factor(1/3) Fractional ideal (3)^-1 sage: K.factor(1+a) Fractional ideal (a + 1) sage: K.factor(1+a/5) (Fractional ideal (-3*a - 2)) * (Fractional ideal (a + 1)) * (Fractional ideal (-a - 2))^-1 * (Fractional ideal (2*a + 1))^-1
Author: Alex Clemesha (2006-05-20): examples
self) |
Return the ideal in
generated by gens. This
overrides the
sage.rings.ring.Field
method to use the
sage.rings.ring.Ring
one instead, since we're not really
concerned with ideals in a field but in its ring of integers.
Input:
sage: K.<a> = NumberField(x^3-2) sage: K.fractional_ideal([1/a]) Fractional ideal (1/2*a^2)
One can also input in a number field ideal itself.
sage: K.fractional_ideal(K.ideal(a)) Fractional ideal (a)
The zero ideal is not a fractional ideal!
sage: K.fractional_ideal(0) Traceback (most recent call last): ... ValueError: gens must have a nonzero element (zero ideal is not a fractional ideal)
self, [pari_group=True], [use_kash=False]) |
Return the Galois group of the Galois closure of this number field as an abstract group.
For more (important!) documentation, so the documentation
for Galois groups of polynomials over
, e.g., by
typing
K.polynomial().galois_group?
, where
is a number field.
To obtain actual field automorphisms that can be applied to
elements, use End(K).list()
and
K.galois_closure()
together (see example below).
sage: k.<b> = NumberField(x^2 - 14) sage: k.galois_group () Galois group PARI group [2, -1, 1, "S2"] of degree 2 of the Number Field in b with defining polynomial x^2 - 14
sage: NumberField(x^3-2, 'a').galois_group(pari_group=True) Galois group PARI group [6, -1, 2, "S3"] of degree 3 of the Number Field in a with defining polynomial x^3 - 2
sage: NumberField(x-1, 'a').galois_group(pari_group=False) # optional database_gap package Galois group Transitive group number 1 of degree 1 of the Number Field in a with defining polynomial x - 1 sage: NumberField(x^2+2, 'a').galois_group(pari_group=False) # optional database_gap package Galois group Transitive group number 1 of degree 2 of the Number Field in a with defining polynomial x^2 + 2 sage: NumberField(x^3-2, 'a').galois_group(pari_group=False) # optional database_gap package Galois group Transitive group number 2 of degree 3 of the Number Field in a with defining polynomial x^3 - 2
EXPLICIT GALOIS GROUP: We compute the Galois group as an explicit group of automorphisms of the Galois closure of a field.
sage: K.<a> = NumberField(x^3 - 2) sage: L.<b1> = K.galois_closure(); L Number Field in b1 with defining polynomial x^6 + 40*x^3 + 1372 sage: G = End(L); G Automorphism group of Number Field in b1 with defining polynomial x^6 + 40*x^3 + 1372 sage: G.list() [ Ring endomorphism of Number Field in b1 with defining polynomial x^6 + 40*x^3 + 1372 Defn: b1 |--> b1, ... Ring endomorphism of Number Field in b1 with defining polynomial x^6 + 40*x^3 + 1372 Defn: b1 |--> -2/63*b1^4 - 31/63*b1 ] sage: G[1](b1) 1/36*b1^4 + 1/18*b1
self, [n=0]) |
Return the generator for this number field.
Input:
sage: k.<theta> = NumberField(x^14 + 2); k Number Field in theta with defining polynomial x^14 + 2 sage: k.gen() theta sage: k.gen(1) Traceback (most recent call last): ... IndexError: Only one generator.
self) |
K.ideal() returns a fractional ideal of the field, except for the zero ideal which is not a fractional ideal.
sage: K.<i>=NumberField(x^2+1) sage: K.ideal(2) Fractional ideal (2) sage: K.ideal(2+i) Fractional ideal (i + 2) sage: K.ideal(0) Ideal (0) of Number Field in i with defining polynomial x^2 + 1
self, bound) |
All integral ideals of bounded norm.
Input:
sage: K.<a> = NumberField(x^2 + 23) sage: d = K.ideals_of_bdd_norm(10) sage: for n in d: ... print n ... for I in d[n]: ... print I 1 Fractional ideal (1) 2 Fractional ideal (2, 1/2*a - 1/2) Fractional ideal (2, 1/2*a + 1/2) 3 Fractional ideal (3, -1/2*a + 1/2) Fractional ideal (3, -1/2*a - 1/2) 4 Fractional ideal (4, 1/2*a + 3/2) Fractional ideal (2) Fractional ideal (4, 1/2*a + 5/2) 5 6 Fractional ideal (-1/2*a + 1/2) Fractional ideal (6, 1/2*a + 5/2) Fractional ideal (6, 1/2*a + 7/2) Fractional ideal (1/2*a + 1/2) 7 8 Fractional ideal (-1/2*a - 3/2) Fractional ideal (4, a - 1) Fractional ideal (4, a + 1) Fractional ideal (1/2*a - 3/2) 9 Fractional ideal (9, 1/2*a + 11/2) Fractional ideal (3) Fractional ideal (9, 1/2*a + 7/2) 10
self, [v=None]) |
Return a list of elements of this number field that are a basis for the full ring of integers.
Input:
sage: K.<a> = NumberField(x^5 + 10*x + 1) sage: K.integral_basis() [1, a, a^2, a^3, a^4]
Next we compute the ring of integers of a cubic field in which 2 is an "essential discriminant divisor", so the ring of integers is not generated by a single element.
sage: K.<a> = NumberField(x^3 + x^2 - 2*x + 8) sage: K.integral_basis() [1, a, 1/2*a^2 + 1/2*a]
self, C) |
Find all integral elements in self with trace in the interval C.
NOTE: This is currently only implemented in the case that self is totally real, since it requires exact computation of self.reduced_gram_matrix().
sage: K.<alpha> = NumberField(ZZ['x'].0^2-2) sage: K.integral_elements_with_trace([0,5]) [alpha + 2, 2, 1] sage: L.<beta> = NumberField(ZZ['x'].0^2+1) sage: L.integral_elements_with_trace([5,11]) Traceback (most recent call last): ... NotImplementedError: exact computation of LLL reduction only implemented in the totally real case
self) |
Return True since a number field is a field.
sage: NumberField(x^5 + x + 3, 'c').is_field() True
self) |
Return True if this number field is a Galois extension of
.
sage: NumberField(x^2 + 1, 'i').is_galois() True sage: NumberField(x^3 + 2, 'a').is_galois() False
self, other) |
Return True if self is isomorphic as a number field to other.
sage: k.<a> = NumberField(x^2 + 1) sage: m.<b> = NumberField(x^2 + 4) sage: k.is_isomorphic(m) True sage: m.<b> = NumberField(x^2 + 5) sage: k.is_isomorphic (m) False
sage: k = NumberField(x^3 + 2, 'a') sage: k.is_isomorphic(NumberField((x+1/3)^3 + 2, 'b')) True sage: k.is_isomorphic(NumberField(x^3 + 4, 'b')) True sage: k.is_isomorphic(NumberField(x^3 + 5, 'b')) False
self) |
sage: K.<a> = NumberField(x^10 - 2) sage: K.is_absolute() True sage: K.is_relative() False
self) |
Return True if self is totally imaginary, and False otherwise.
Totally imaginary means that no isomorphic embedding of self into the complex numbers has image contained in the real numbers.
sage: NumberField(x^2+2, 'alpha').is_totally_imaginary() True sage: NumberField(x^2-2, 'alpha').is_totally_imaginary() False sage: NumberField(x^4-2, 'alpha').is_totally_imaginary() False
self) |
Return True if self is totally real, and False otherwise.
Totally real means that every isomorphic embedding of self into the complex numbers has image contained in the real numbers.
sage: NumberField(x^2+2, 'alpha').is_totally_real() False sage: NumberField(x^2-2, 'alpha').is_totally_real() True sage: NumberField(x^4-2, 'alpha').is_totally_real() False
self, [name=None]) |
Return the latex representation of the variable name for this number field.
sage: NumberField(x^2 + 3, 'a').latex_variable_name() 'a' sage: NumberField(x^3 + 3, 'theta3').latex_variable_name() '\theta_{3}' sage: CyclotomicField(5).latex_variable_name() '\zeta_{5}'
self, [proof=None]) |
Return the narrow class group of this field.
Input:
sage: NumberField(x^3+x+9, 'a').narrow_class_group() Multiplicative Abelian Group isomorphic to C2
self) |
Return the number of generators of this number field (always 1).
Output: the python integer 1.
sage: NumberField(x^2 + 17,'a').ngens() 1 sage: NumberField(x + 3,'a').ngens() 1 sage: k.<a> = NumberField(x + 3) sage: k.ngens() 1 sage: k.0 -3
self) |
Return number of roots of unity in this field.
sage: K.<b> = NumberField(x^2+1) sage: K.number_of_roots_of_unity() 4
self) |
Return the order of this number field (always +infinity).
Output: always positive infinity
sage: NumberField(x^2 + 19,'a').order() +Infinity
self, [certify=False], [units=True]) |
PARI big number field corresponding to this field.
sage: k.<a> = NumberField(x^2 + 1); k Number Field in a with defining polynomial x^2 + 1 sage: len(k.pari_bnf()) 10 sage: k.pari_bnf()[:4] [[;], matrix(0,7), [;], ...] sage: len(k.pari_nf()) 9
self) |
Run the PARI bnfcertify function to ensure the correctness of answers.
If this function returns True (and doesn't raise a ValueError), then certification succeeded, and results that use the PARI bnf structure with this field are supposed to be correct.
WARNING: I wouldn't trust this to mean that everything computed involving this number field is actually correct.
sage: k.<a> = NumberField(x^7 + 7); k Number Field in a with defining polynomial x^7 + 7 sage: k.pari_bnf_certify() True
self) |
PARI number field corresponding to this field.
This is the number field constructed using nfinit. This is the same as the number field got by doing pari(self) or gp(self).
sage: k.<a> = NumberField(x^4 - 3*x + 7); k Number Field in a with defining polynomial x^4 - 3*x + 7 sage: k.pari_nf()[:4] [x^4 - 3*x + 7, [0, 2], 85621, 1] sage: pari(k)[:4] [x^4 - 3*x + 7, [0, 2], 85621, 1]
sage: k.<a> = NumberField(x^4 - 3/2*x + 5/3); k Number Field in a with defining polynomial x^4 - 3/2*x + 5/3 sage: k.pari_nf() Traceback (most recent call last): ... TypeError: Unable to coerce number field defined by non-integral polynomial to PARI. sage: pari(k) Traceback (most recent call last): ... TypeError: Unable to coerce number field defined by non-integral polynomial to PARI. sage: gp(k) Traceback (most recent call last): ... TypeError: Unable to coerce number field defined by non-integral polynomial to PARI.
self, [name=x]) |
PARI polynomial corresponding to polynomial that defines this field. By default, this is a polynomial in the variable "x".
sage: y = polygen(QQ) sage: k.<a> = NumberField(y^2 - 3/2*y + 5/3) sage: k.pari_polynomial() x^2 - 3/2*x + 5/3 sage: k.pari_polynomial('a') a^2 - 3/2*a + 5/3
self) |
Return the defining polynomial of this number field.
This is exactly the same as self.defining_polynomal()
.
sage: NumberField(x^2 + (2/3)*x - 9/17,'a').polynomial() x^2 + 2/3*x - 9/17
self) |
Return defining polynomial of this number field as a pair, an ntl polynomial and a denominator.
This is used mainly to implement some internal arithmetic.
sage: NumberField(x^2 + (2/3)*x - 9/17,'a').polynomial_ntl() ([-27 34 51], 51)
self) |
Return the polynomial quotient ring isomorphic to this number field.
sage: K = NumberField(x^3 + 2*x - 5, 'alpha') sage: K.polynomial_quotient_ring() Univariate Quotient Polynomial Ring in alpha over Rational Field with modulus x^3 + 2*x - 5
self) |
Return the polynomial ring that we view this number field as being a quotient of (by a principal ideal).
An example with an absolute field:
sage: k.<a> = NumberField(x^2 + 3) sage: y = polygen(QQ, 'y') sage: k.<a> = NumberField(y^2 + 3) sage: k.polynomial_ring() Univariate Polynomial Ring in y over Rational Field
An example with a relative field:
sage: y = polygen(QQ, 'y') sage: M.<a> = NumberField([y^3 + 97, y^2 + 1]); M Number Field in a0 with defining polynomial y^3 + 97 over its base field sage: M.polynomial_ring() Univariate Polynomial Ring in y over Number Field in a1 with defining polynomial y^2 + 1
self, x, [degree=None]) |
Return a prime ideal of self lying over x.
Input:
WARNING: at this time we factor the ideal x, which may not be supported for relative number fields.
sage: x = ZZ['x'].gen() sage: F.<t> = NumberField(x^3 - 2)
sage: P2 = F.prime_above(2) sage: P2 # random Fractional ideal (-t) sage: 2 in P2 True sage: P2.is_prime() True sage: P2.norm() 2
sage: P3 = F.prime_above(3) sage: P3 # random Fractional ideal (t + 1) sage: 3 in P3 True sage: P3.is_prime() True sage: P3.norm() 3
The ideal (3) is totally ramified in F, so there is no degree 2 prime above 3:
sage: F.prime_above(3, degree=2) Traceback (most recent call last): ... ValueError: No prime of degree 2 above Fractional ideal (3) sage: [ id.residue_class_degree() for id, _ in F.ideal(3).factor() ] [1]
Asking for a specific degree works:
sage: P5_1 = F.prime_above(5, degree=1) sage: P5_1 # random Fractional ideal (-t^2 - 1) sage: P5_1.residue_class_degree() 1
sage: P5_2 = F.prime_above(5, degree=2) sage: P5_2 # random Fractional ideal (t^2 - 2*t - 1) sage: P5_2.residue_class_degree() 2
TESTS: It doesn't make sense to factor the ideal (0):
sage: F.prime_above(0) Traceback (most recent call last): ... AttributeError: 'NumberFieldIdeal' object has no attribute 'factor'
Sage can't factor ideals over extension fields yet:
sage: G = F.extension(x^2 - 11, 'b') sage: G.prime_above(13) Traceback (most recent call last): ... NotImplementedError
self, x, [degree=None]) |
Return prime ideals of self lying over x.
Input:
WARNING: at this time we factor the ideal x, which may not be supported for relative number fields.
sage: x = ZZ['x'].gen() sage: F.<t> = NumberField(x^3 - 2)
sage: P2s = F.primes_above(2) sage: P2s # random [Fractional ideal (-t)] sage: all(2 in P2 for P2 in P2s) True sage: all(P2.is_prime() for P2 in P2s) True sage: [ P2.norm() for P2 in P2s ] [2]
sage: P3s = F.primes_above(3) sage: P3s # random [Fractional ideal (t + 1)] sage: all(3 in P3 for P3 in P3s) True sage: all(P3.is_prime() for P3 in P3s) True sage: [ P3.norm() for P3 in P3s ] [3]
The ideal (3) is totally ramified in F, so there is no degree 2 prime above 3:
sage: F.primes_above(3, degree=2) [] sage: [ id.residue_class_degree() for id, _ in F.ideal(3).factor() ] [1]
Asking for a specific degree works:
sage: P5_1s = F.primes_above(5, degree=1) sage: P5_1s # random [Fractional ideal (-t^2 - 1)] sage: P5_1 = P5_1s[0]; P5_1.residue_class_degree() 1
sage: P5_2s = F.primes_above(5, degree=2) sage: P5_2s # random [Fractional ideal (t^2 - 2*t - 1)] sage: P5_2 = P5_2s[0]; P5_2.residue_class_degree() 2
TESTS: It doesn't make sense to factor the ideal (0):
sage: F.primes_above(0) Traceback (most recent call last): ... AttributeError: 'NumberFieldIdeal' object has no attribute 'factor'
Sage can't factor ideals over extension fields yet:
sage: G = F.extension(x^2 - 11, 'b') sage: G.primes_above(13) Traceback (most recent call last): ... NotImplementedError
self, [num_integer_primes=10000], [max_iterations=100]) |
Return an iterator yielding prime ideals of absolute degree one and small norm.
WARNING:
It is possible that there are no primes of
of absolute degree
one of small prime norm, and it possible that this algorithm will
not find any primes of small norm.
See module sage.rings.number_field.small_primes_of_degree_one for details.
Input:
sage: K.<z> = CyclotomicField(10) sage: it = K.primes_of_degree_one_iter() sage: Ps = [ it.next() for i in range(3) ] sage: Ps # random [Fractional ideal (z^3 + z + 1), Fractional ideal (3*z^3 - z^2 + z - 1), Fractional ideal (2*z^3 - 3*z^2 + z - 2)] sage: [ P.norm() for P in Ps ] # random [11, 31, 41] sage: [ P.residue_class_degree() for P in Ps ] [1, 1, 1]
self, n, [num_integer_primes=10000], [max_iterations=100]) |
Return a list of n prime ideals of absolute degree one and small norm.
WARNING:
It is possible that there are no primes of
of absolute degree
one of small prime norm, and it possible that this algorithm will
not find any primes of small norm.
See module sage.rings.number_field.small_primes_of_degree_one for details.
Input:
sage: K.<z> = CyclotomicField(10) sage: Ps = K.primes_of_degree_one_list(3) sage: Ps [Fractional ideal (z^3 + z + 1), Fractional ideal (3*z^3 - z^2 + z - 1), Fractional ideal (2*z^3 - 3*z^2 + z - 2)] sage: [ P.norm() for P in Ps ] [11, 31, 41] sage: [ P.residue_class_degree() for P in Ps ] [1, 1, 1]
self) |
Return a primitive element for this field, i.e., an element
that generates it over
.
sage: K.<a> = NumberField(x^3 + 2) sage: K.primitive_element() a sage: K.<a,b,c> = NumberField([x^2-2,x^2-3,x^2-5]) sage: K.primitive_element() a + (-1)*b + c sage: alpha = K.primitive_element(); alpha a + (-1)*b + c sage: alpha.minpoly() x^2 + (2*b - 2*c)*x + (-2*c)*b + 6 sage: alpha.absolute_minpoly() x^8 - 40*x^6 + 352*x^4 - 960*x^2 + 576
self, [prec=53]) |
Return all homomorphisms of this number field into the approximate real field with precision prec.
If prec is 53 (the default), then the real double field is used; otherwise the arbitrary precision (but slow) real field is used.
sage: K.<a> = NumberField(x^3 + 2) sage: K.real_embeddings() [ Ring morphism: From: Number Field in a with defining polynomial x^3 + 2 To: Real Double Field Defn: a |--> -1.25992104989 ] sage: K.real_embeddings(16) [ Ring morphism: From: Number Field in a with defining polynomial x^3 + 2 To: Real Field with 16 bits of precision Defn: a |--> -1.260 ] sage: K.real_embeddings(100) [ Ring morphism: From: Number Field in a with defining polynomial x^3 + 2 To: Real Field with 100 bits of precision Defn: a |--> -1.2599210498948731647672106073 ]
self, [prec=None]) |
This function returns an LLL-reduced basis for the Minkowski-embedding of the maximal order of a number field.
Input:
NOTE: In the non-totally-real case, the LLL routine we call is currently Pari's qflll(), which works with floating point approximations, and so the result is only as good as the precision promised by Pari. The matrix returned will always be integral; however, it may only be only "almost" LLL-reduced when the precision is not sufficiently high.
sage: F.<t> = NumberField(x^6-7*x^4-x^3+11*x^2+x-1) sage: F.maximal_order().basis() [1/2*t^5 + 1/2*t^4 + 1/2*t^2 + 1/2, t, t^2, t^3, t^4, t^5] sage: F.reduced_basis() [1, 1/2*t^5 - 1/2*t^4 - 3*t^3 + 3/2*t^2 + 4*t - 1/2, t, 1/2*t^5 + 1/2*t^4 - 4*t^3 - 5/2*t^2 + 7*t + 1/2, 1/2*t^5 - 1/2*t^4 - 2*t^3 + 3/2*t^2 - 1/2, 1/2*t^5 - 1/2*t^4 - 3*t^3 + 5/2*t^2 + 4*t - 5/2]
sage: F.<alpha> = NumberField(x^4+x^2+712312*x+131001238) sage: F.integral_basis() [1, alpha, alpha^2, 1/2*alpha^3 + 1/2*alpha^2] sage: F.reduced_basis(prec=10) [1, alpha, alpha^2 - 15*alpha + 8, alpha^3 - 16*alpha^2 + 471*alpha + 266719] sage: F.reduced_basis(prec=300) [1, alpha, alpha^2 - 15*alpha, alpha^3 - 16*alpha^2 + 469*alpha + 267109]
self, [prec=None]) |
This function returns the Gram matrix of an LLL-reduced basis for the Minkowski embedding of the maximal order of a number field.
Input:
NOTE: In the non-totally-real case, the LLL routine we call is currently Pari's qflll(), which works with floating point approximations, and so the result is only as good as the precision promised by Pari. In particular, in this case, the returned matrix will *not* be integral, and may not have enough precision to recover the correct gram matrix (which is known to be integral for theoretical reasons). Thus the need for the prec flag above.
sage: F.<t> = NumberField(x^6-7*x^4-x^3+11*x^2+x-1) sage: F.reduced_gram_matrix() [ 6 -3 0 -2 0 -1] [-3 9 0 1 0 3] [ 0 0 14 6 -2 3] [-2 1 6 16 -3 3] [ 0 0 -2 -3 16 6] [-1 3 3 3 6 19] sage: Matrix(6, [(x*y).trace() for x in F.integral_basis() for y in F.integral_basis()]) [ 6 0 14 3 54 52] [ 0 14 3 54 30 133] [ 14 3 54 30 233 259] [ 3 54 30 233 217 664] [ 54 30 233 217 1078 1368] [ 52 133 259 664 1368 2550]
sage: var('x') x sage: F.<alpha> = NumberField(x^4+x^2+712312*x+131001238) sage: F.reduced_gram_matrix() # random low-order bits [ 3.99999999998249 0.000000000000000 1.99999999997817 -1.06846799999532e6] [ 0.000000000000000 46721.5393313587 11488.9100265019 -1.12285582008158e7] [ 1.99999999997817 11488.9100265019 5.56589153102570e8 8.06191790906345e9] [-1.06846799999532e6 -1.12285582008158e7 8.06191790906345e9 5.87118790062408e12]
self, [proof=None]) |
Return the regulator of this number field.
Note that PARI computes the regulator to higher precision than the SAGE default.
Input:
sage: NumberField(x^2-2, 'a').regulator() 0.88137358701954305 sage: NumberField(x^4+x^3+x^2+x+1, 'a').regulator() 0.96242365011920694
self, prime, [names=None], [check=False]) |
Return the residue field of this number field at a given prime, ie
.
Input:
sage: R.<x> = QQ[] sage: K.<a> = NumberField(x^4+3*x^2-17) sage: P = K.ideal(61).factor()[0][0] sage: K.residue_field(P) Residue field in abar of Fractional ideal (-2*a^2 + 1)
self) |
Return all the roots of unity in this field, primitive or not.
sage: K.<b> = NumberField(x^2+1) sage: zs = K.roots_of_unity(); zs [b, -1, -b, 1] sage: [ z**K.number_of_roots_of_unity() for z in zs ] [1, 1, 1, 1]
self) |
Return (r1, r2), where r1 and r2 are the number of real embeddings and pairs of complex embeddings of this field, respectively.
sage: NumberField(x^2+1, 'a').signature() (0, 1) sage: NumberField(x^3-2, 'a').signature() (1, 1)
self) |
Return fixed isomorphism or embedding structure on self.
This is used to record various isomorphisms or embeddings that arise naturally in other constructions.
sage: K.<z> = NumberField(x^2 + 3) sage: L.<a> = K.absolute_field(); L Number Field in a with defining polynomial x^2 + 3 sage: L.structure() (Number field isomorphism from Number Field in a with defining polynomial x^2 + 3 to Number Field in z with defining polynomial x^2 + 3 given by variable name change, Number field isomorphism from Number Field in z with defining polynomial x^2 + 3 to Number Field in a with defining polynomial x^2 + 3 given by variable name change)
self, alpha, [name=None]) |
Return an absolute number field K isomorphic to QQ(alpha) and a map from K to self that sends the generator of K to alpha.
Input:
sage: K.<a> = NumberField(x^4 - 3); K Number Field in a with defining polynomial x^4 - 3 sage: H, from_H = K.subfield(a^2, name='b') sage: H Number Field in b with defining polynomial x^2 - 3 sage: from_H(H.0) a^2 sage: from_H Ring morphism: From: Number Field in b with defining polynomial x^2 - 3 To: Number Field in a with defining polynomial x^4 - 3 Defn: b |--> a^2
sage: K.<z> = CyclotomicField(5) sage: K.subfield(z-z^2-z^3+z^4) (Number Field in z0 with defining polynomial x^2 - 5, Ring morphism: From: Number Field in z0 with defining polynomial x^2 - 5 To: Cyclotomic Field of order 5 and degree 4 Defn: z0 |--> -2*z^3 - 2*z^2 - 1)
You can also view a number field as having a different
generator by just choosing the input to generate the
whole field; for that it is better to use
self.change_generator
, which gives isomorphisms
in both directions.
self, v) |
Return the matrix of the trace pairing on the elements of the
list
.
sage: K.<zeta3> = NumberField(x^2 + 3) sage: K.trace_pairing([1,zeta3]) [ 2 0] [ 0 -6]
self, P, [others=positive]) |
Returns an element of self with valuation 1 at the prime ideal P.
Input:
NOTE: When P is principal (e.g. always when self has class number one) the result may or may not be a generator of P!
sage: K.<a> = NumberField(x^2 + 5); K Number Field in a with defining polynomial x^2 + 5 sage: P,Q = K.ideal(3).prime_factors() sage: P Fractional ideal (3, a + 1) sage: pi=K.uniformizer(P); pi a + 1 sage: K.ideal(pi).factor() (Fractional ideal (2, a + 1)) * (Fractional ideal (3, a + 1)) sage: pi=K.uniformizer(P,'negative'); pi 1/2*a + 1/2 sage: K.ideal(pi).factor() (Fractional ideal (2, a + 1))^-1 * (Fractional ideal (3, a + 1))
sage: K = CyclotomicField(9) sage: Plist=K.ideal(17).prime_factors() sage: pilist = [K.uniformizer(P) for P in Plist] sage: [pi.is_integral() for pi in pilist] [True, True, True] sage: [pi.valuation(P) for pi,P in zip(pilist,Plist)] [1, 1, 1] sage: [ pilist[i] in Plist[i] for i in range(len(Plist)) ] [True, True, True]
self, [proof=None]) |
Return generators for the unit group modulo torsion.
ALGORITHM: Uses PARI's bnfunit command.
INPUTS: proof - default: True
sage: x = QQ['x'].0 sage: A = x^4 - 10*x^3 + 20*5*x^2 - 15*5^2*x + 11*5^3 sage: K = NumberField(A, 'a') sage: K.units() [8/275*a^3 - 12/55*a^2 + 15/11*a - 2]
Sage might not be able to provably compute the unit group:
sage: K = NumberField(x^17 + 3, 'a') sage: K.units(proof=True) # default Traceback (most recent call last): ... PariError: not enough precomputed primes, need primelimit ~ (35)
In this case, one can ask for the conjectural unit group (correct if the Generalized Riemann Hypothesis is true):
sage: K.units(proof=False) [a^9 + a - 1, a^16 - a^15 + a^14 - a^12 + a^11 - a^10 - a^8 + a^7 - 2*a^6 + a^4 - 3*a^3 + 2*a^2 - 2*a + 1, 2*a^16 - a^14 - a^13 + 3*a^12 - 2*a^10 + a^9 + 3*a^8 - 3*a^6 + 3*a^5 + 3*a^4 - 2*a^3 - 2*a^2 + 3*a + 4, 2*a^16 - 3*a^15 + 3*a^14 - 3*a^13 + 3*a^12 - a^11 + a^9 - 3*a^8 + 4*a^7 - 5*a^6 + 6*a^5 - 4*a^4 + 3*a^3 - 2*a^2 - 2*a + 4, a^15 - a^12 + a^10 - a^9 - 2*a^8 + 3*a^7 + a^6 - 3*a^5 + a^4 + 4*a^3 - 3*a^2 - 2*a + 2, a^16 - a^15 - 3*a^14 - 4*a^13 - 4*a^12 - 3*a^11 - a^10 + 2*a^9 + 4*a^8 + 5*a^7 + 4*a^6 + 2*a^5 - 2*a^4 - 6*a^3 - 9*a^2 - 9*a - 7, a^15 + a^14 + 2*a^11 + a^10 - a^9 + a^8 + 2*a^7 - a^5 + 2*a^3 - a^2 - 3*a + 1, 3*a^16 + 3*a^15 + 3*a^14 + 3*a^13 + 3*a^12 + 2*a^11 + 2*a^10 + 2*a^9 + a^8 - a^7 - 2*a^6 - 3*a^5 - 3*a^4 - 4*a^3 - 6*a^2 - 8*a - 8]
The provable and the conjectural results are cached separately (this fixes trac #2504):
sage: K.units(proof=True) Traceback (most recent call last): ... PariError: not enough precomputed primes, need primelimit ~ (35)
self, [n=2], [all=False]) |
If all is False, return a primitive n-th root of unity in this field, or raise an ArithmeticError exception if there are none.
If all is True, return a list of all primitive n-th roots of unity in this field (possibly empty).
Note that if one wants to know the maximal root of unity in this field, one can use self.zeta_order().
Input:
sage: K.<z> = NumberField(x^2 + 3) sage: K.zeta(1) 1 sage: K.zeta(2) -1 sage: K.zeta(2, all=True) [-1] sage: K.zeta(3) 1/2*z - 1/2 sage: K.zeta(3, all=True) [1/2*z - 1/2, -1/2*z - 1/2] sage: K.zeta(4) Traceback (most recent call last): ... ArithmeticError: There are no 4-th roots of unity in self.
sage: r.<x> = QQ[] sage: K.<b> = NumberField(x^2+1) sage: K.zeta(4) b sage: K.zeta(4,all=True) [b, -b] sage: K.zeta(3) Traceback (most recent call last): ... ArithmeticError: There are no 3-rd roots of unity in self. sage: K.zeta(3,all=True) []
self, n) |
Compute the first n coefficients of the Dedekind zeta function of this field as a Dirichlet series.
sage: x = QQ['x'].0 sage: NumberField(x^2+1, 'a').zeta_coefficients(10) [1, 1, 0, 1, 2, 0, 0, 1, 1, 2]
self, [prec=53], [max_imaginary_part=0], [max_asymp_coeffs=40]) |
Return the Zeta function of this number field.
This actually returns an interface to Tim Dokchitser's program for computing with the Dedekind zeta function zeta_F(s) of the number field F.
Input:
sage: K.<a> = NumberField(ZZ['x'].0^2+ZZ['x'].0-1) sage: Z = K.zeta_function() sage: Z Zeta function associated to Number Field in a with defining polynomial x^2 + x - 1 sage: Z(-1) 0.0333333333333333
self) |
Return the number of roots of unity in this field.
sage: F.<alpha> = NumberField(x**22+3) sage: F.zeta_order() 6 sage: F.<alpha> = NumberField(x**2-7) sage: F.zeta_order() 2
Special Functions: __call__,
__cmp__,
__init__,
_coerce_from_other_number_field,
_coerce_from_str,
_coerce_impl,
_coerce_non_number_field_element_in,
_fractional_ideal_class_,
_gap_init_,
_Hom_,
_ideal_class_,
_is_valid_homomorphism_,
_latex_,
_normalize_prime_list,
_pari_init_,
_repr_,
_set_structure
self, x) |
Coerce x into this number field.
sage: K.<a> = NumberField(x^3 + 17) sage: K(a) is a True sage: K('a^2 + 2/3*a + 5') a^2 + 2/3*a + 5 sage: K('1').parent() Number Field in a with defining polynomial x^3 + 17 sage: K(3/5).parent() Number Field in a with defining polynomial x^3 + 17
self, other) |
Compare a number field with something else.
Input:
If other is not a number field, then the types of self and other are compared. If both are number fields, then the variable names are compared. If those are the same, then the underlying defining polynomials are compared. If the polynomials are the same, the number fields are considered ``equal'', but need not be identical. Coercion between equal number fields is allowed.
sage: k.<a> = NumberField(x^3 + 2); m.<b> = NumberField(x^3 + 2) sage: cmp(k,m) -1 sage: cmp(m,k) 1 sage: k == QQ False sage: k.<a> = NumberField(x^3 + 2); m.<a> = NumberField(x^3 + 2) sage: k is m True sage: m = loads(dumps(k)) sage: k is m False sage: k == m True
self, x) |
Coerce a number field element x into this number field.
In most cases this currently doesn't work (since it is barely implemented) - it only works for constants.
Input:
sage: K.<a> = NumberField(x^3 + 2) sage: L.<b> = NumberField(x^2 + 1) sage: K._coerce_from_other_number_field(L(2/3)) 2/3
self, x) |
Coerce a string representation of an element of this number field into this number field.
Input:
sage: k.<theta25> = NumberField(x^3+(2/3)*x+1) sage: k._coerce_from_str('theta25^3 + (1/3)*theta25') -1/3*theta25 - 1
This function is called by the coerce method when it gets a string as input:
sage: k('theta25^3 + (1/3)*theta25') -1/3*theta25 - 1
self, x) |
Canonical coercion of x into self.
Currently integers, rationals, and this field itself coerce canonical into this field.
sage: S.<y> = NumberField(x^3 + x + 1) sage: S._coerce_impl(int(4)) 4 sage: S._coerce_impl(long(7)) 7 sage: S._coerce_impl(-Integer(2)) -2 sage: z = S._coerce_impl(-7/8); z, type(z) (-7/8, <type 'sage.rings.number_field.number_field_element.NumberFieldEleme nt_absolute'>) sage: S._coerce_impl(y) is y True
There are situations for which one might imagine canonical coercion could make sense (at least after fixing choices), but which aren't yet implemented:
sage: K.<a> = QuadraticField(2) sage: K._coerce_impl(sqrt(2)) Traceback (most recent call last): ... TypeError
self, x) |
Coerce a non-number field element x into this number field.
Input:
sage: K.<a> = NumberField(x^3 + 2/3) sage: K._coerce_non_number_field_element_in(-7/8) -7/8 sage: K._coerce_non_number_field_element_in([1,2,3]) 3*a^2 + 2*a + 1
The list is just turned into a polynomial in the generator.
sage: K._coerce_non_number_field_element_in([0,0,0,1,1]) -2/3*a - 2/3
Not any polynomial coerces in, e.g., not this one in characteristic 7.
sage: f = GF(7)['y']([1,2,3]); f 3*y^2 + 2*y + 1 sage: K._coerce_non_number_field_element_in(f) Traceback (most recent call last): ... TypeError
self) |
Return the Python class used in defining fractional ideals of the ring of integers of this number field.
This function is required by the general ring/ideal machinery. The value defined here is the default value for all number fields *except* relative number fields; this function is overridden by one of the same name on class NumberField_relative.
sage: NumberField(x^2 + 2, 'c')._fractional_ideal_class_() <class 'sage.rings.number_field.number_field_ideal.NumberFieldFractionalIde al'>
self) |
Create a gap object representing self and return its name
sage: F=CyclotomicField(8) sage: F.gen() zeta8 sage: F._gap_init_() # the following variable name $sage1 represents the F.base_ring() in gap and is somehow random 'CallFuncList(function() local x,E; x:=Indeterminate($sage1,"x"); E:=AlgebraicExtension($sage1,x^4 + 1,"zeta8"); return E; end,[])' sage: f=gap(F) sage: f <algebraic extension over the Rationals of degree 4> sage: f.GeneratorsOfDivisionRing() [ (zeta8) ]
self, codomain, [cat=None]) |
Return homset of homomorphisms from self to the number field codomain.
The cat option is currently ignored.
This function is implicitly called by the Hom method or function.
sage: K.<i> = NumberField(x^2 + 1); K Number Field in i with defining polynomial x^2 + 1 sage: K.Hom(K) Automorphism group of Number Field in i with defining polynomial x^2 + 1 sage: Hom(K, QuadraticField(-1, 'b')) Set of field embeddings from Number Field in i with defining polynomial x^2 + 1 to Number Field in b with defining polynomial x^2 + 1
self) |
Return the Python class used in defining the zero ideal of the ring of integers of this number field.
This function is required by the general ring/ideal machinery. The value defined here is the default value for all number fields.
sage: NumberField(x^2 + 2, 'c')._ideal_class_() <class 'sage.rings.number_field.number_field_ideal.NumberFieldIdeal'>
self, codomain, im_gens) |
Return whether or not there is a homomorphism defined by the given images of generators.
To do this we just check that the elements of the image of the given generator (im_gens always has length 1) satisfies the relation of the defining poly of this field.
sage: k.<a> = NumberField(x^2 - 3) sage: k._is_valid_homomorphism_(QQ, [0]) False sage: k._is_valid_homomorphism_(k, []) False sage: k._is_valid_homomorphism_(k, [a]) True sage: k._is_valid_homomorphism_(k, [-a]) True sage: k._is_valid_homomorphism_(k, [a+1]) False
self) |
Return latex representation of this number field. This is viewed as a polynomial quotient ring over a field.
sage: k.<a> = NumberField(x^13 - (2/3)*x + 3) sage: k._latex_() '\\mathbf{Q}[a]/(a^{13} - \\frac{2}{3} a + 3)' sage: latex(k) \mathbf{Q}[a]/(a^{13} - \frac{2}{3} a + 3)
Numbered variables are often correctly typeset:
sage: k.<theta25> = NumberField(x^25+x+1) sage: print k._latex_() \mathbf{Q}[\theta_{25}]/(\theta_{25}^{25} + \theta_{25} + 1)
self) |
Needed for conversion of number field to PARI.
This only works if the defining polynomial of this number field is integral and monic.
sage: k = NumberField(x^2 + x + 1, 'a') sage: k._pari_init_() 'nfinit(x^2 + x + 1)' sage: k._pari_() [x^2 + x + 1, [0, 1], -3, 1, ... [1, x], [1, 0; 0, 1], [1, 0, 0, -1; 0, 1, 1, -1]] sage: pari(k) [x^2 + x + 1, [0, 1], -3, 1, ...[1, x], [1, 0; 0, 1], [1, 0, 0, -1; 0, 1, 1, -1]]
self) |
Return string representation of this number field.
sage: k.<a> = NumberField(x^13 - (2/3)*x + 3) sage: k._repr_() 'Number Field in a with defining polynomial x^13 - 2/3*x + 3'
Class: NumberField_quadratic
The command QuadraticField(a) creates the field Q(sqrt(a)).
sage: QuadraticField(3, 'a') Number Field in a with defining polynomial x^2 - 3 sage: QuadraticField(-4, 'b') Number Field in b with defining polynomial x^2 + 4
self, polynomial, [name=None], [check=True]) |
Create a quadratic number field.
sage: k.<a> = QuadraticField(5, check=False); k Number Field in a with defining polynomial x^2 - 5
Don't do this:
sage: k.<a> = QuadraticField(4, check=False); k Number Field in a with defining polynomial x^2 - 4
Functions: class_number,
coerce_map_from_impl,
discriminant,
hilbert_class_field,
hilbert_class_polynomial,
is_galois
self, [proof=None]) |
Return the size of the class group of self.
If proof = False (*not* the default!) and the discriminant of the
field is negative, then the following warning from the PARI
manual applies: IMPORTANT WARNING: For
, this function
may give incorrect results when the class group has a low
exponent (has many cyclic factors), because implementing
Shank's method in full generality slows it down immensely.
sage: QuadraticField(-23,'a').class_number() 3
These are all the primes so that the class number of
is
:
sage: [d for d in prime_range(2,300) if not is_square(d) and QuadraticField(-d,'a').class_number() == 1] [2, 3, 7, 11, 19, 43, 67, 163]
It is an open problem to prove that there are infinity
many positive square-free
such that
has
class number
:n
sage: len([d for d in range(2,200) if not is_square(d) and QuadraticField(d,'a').class_number() == 1]) 121
self, S) |
sage: K.<a> = QuadraticField(-3) sage: f = K.coerce_map_from(QQ); f Natural morphism: From: Rational Field To: Number Field in a with defining polynomial x^2 + 3 sage: f(3/5) 3/5 sage: parent(f(3/5)) is K True
self, [v=None]) |
Returns the discriminant of the ring of integers of the number field, or if v is specified, the determinant of the trace pairing on the elements of the list v.
Input:
sage: K.<i> = NumberField(x^2+1) sage: K.discriminant() -4 sage: K.<a> = NumberField(x^2+5) sage: K.discriminant() -20 sage: K.<a> = NumberField(x^2-5) sage: K.discriminant() 5
self, names) |
Returns the Hilbert class field of this quadratic field as an
absolute extension of
. For a polynomial that defines a
relative extension see the
hilbert_class_polynomial
command.
Note: Computed using PARI via Schertz's method. This implementation is amazingly fast.
sage: x = QQ['x'].0 sage: K = NumberField(x^2 + 23, 'a') sage: K.hilbert_class_polynomial() x^3 + x^2 - 1 sage: K.hilbert_class_field('h') Number Field in h with defining polynomial x^6 + 2*x^5 + 70*x^4 + 90*x^3 + 1631*x^2 + 1196*x + 12743
self) |
Returns a polynomial over
whose roots generate the
Hilbert class field of this quadratic field.
Note: Computed using PARI via Schertz's method. This implementation is quite fast.
sage: K.<b> = QuadraticField(-23) sage: K.hilbert_class_polynomial() x^3 + x^2 - 1
sage: K.<a> = QuadraticField(-431) sage: K.class_number() 21 sage: K.hilbert_class_polynomial() x^21 + x^20 - 13*x^19 - 50*x^18 + 592*x^17 - 2403*x^16 + 5969*x^15 - 10327*x^14 + 13253*x^13 - 12977*x^12 + 9066*x^11 - 2248*x^10 - 5523*x^9 + 11541*x^8 - 13570*x^7 + 11315*x^6 - 6750*x^5 + 2688*x^4 - 577*x^3 + 9*x^2 + 15*x + 1
self) |
Return True since all quadratic fields are automatically Galois.
sage: QuadraticField(1234,'d').is_galois() True
Special Functions: __init__,
__reduce__
self) |
This is used in pickling quadratic number fields.
TESTS:
sage: K.<z7> = QuadraticField(7) sage: L = loads(dumps(K)) sage: print L Number Field in z7 with defining polynomial x^2 - 7 sage: print L == K True
Class: NumberField_relative
sage: K.<a> = NumberField(x^3 - 2) sage: t = K['x'].gen() sage: L.<b> = K.extension(t^2+t+a); L Number Field in b with defining polynomial x^2 + x + a over its base field
self, base, polynomial, name, [latex_name=None], [names=None], [check=True]) |
Input:
K['x']
, where
K is the base field.
sage: K.<x> = CyclotomicField(5)[] sage: W.<a> = NumberField(x^2 + 1) sage: W Number Field in a with defining polynomial x^2 + 1 over its base field sage: type(W) <class 'sage.rings.number_field.number_field.NumberField_relative'>
Test that check=False really skips the test:
sage: W.<a> = NumberField(K.cyclotomic_polynomial(5), check=False) sage: W Number Field in a with defining polynomial x^4 + x^3 + x^2 + x + 1 over its base field
A relative extension of a relative extension:
sage: x = var('x') sage: k.<a> = NumberField([x^2 + 2, x^2 + 1]) sage: l.<b> = k.extension(x^2 + 3) sage: l Number Field in b with defining polynomial x^2 + 3 over its base field sage: l.base_field() Number Field in a0 with defining polynomial x^2 + 2 over its base field sage: l.base_field().base_field() Number Field in a1 with defining polynomial x^2 + 1
Functions: absolute_base_field,
absolute_degree,
absolute_field,
absolute_generator,
absolute_polynomial,
absolute_polynomial_ntl,
absolute_vector_space,
base_field,
base_ring,
change_names,
embeddings,
galois_closure,
galois_group,
gen,
gens,
is_absolute,
is_free,
is_galois,
lift_to_base,
maximal_order,
ngens,
number_of_roots_of_unity,
order,
pari_polynomial,
pari_relative_polynomial,
pari_rnf,
polynomial,
relative_discriminant,
relativize,
roots_of_unity,
vector_space
self) |
Return the base field of this relative extension, but viewed as an absolute field over QQ.
sage: K.<a,b,c> = NumberField([x^2 + 2, x^3 + 3, x^3 + 2]) sage: K Number Field in a with defining polynomial x^2 + 2 over its base field sage: K.base_field() Number Field in b with defining polynomial x^3 + 3 over its base field sage: K.absolute_base_field()[0] Number Field in a with defining polynomial x^9 + 3*x^6 + 165*x^3 + 1 sage: K.base_field().absolute_field('z') Number Field in z with defining polynomial x^9 + 3*x^6 + 165*x^3 + 1
self) |
sage: K.<a> = NumberField([x^2 + 3, x^2 + 2]) sage: K.absolute_degree() 4 sage: K.degree() 2
self, names) |
Return an absolute number field K that is isomorphic to this field along with a field-theoretic bijection from self to K and from K to self.
Input:
Also, K.structure()
returns from_K and to_K, where
from_K is an isomorphism from K to self and to_K is an isomorphism
from self to K.
sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K Number Field in a with defining polynomial x^4 + 3 over its base field sage: L.<xyz> = K.absolute_field(); L Number Field in xyz with defining polynomial x^8 + 8*x^6 + 30*x^4 - 40*x^2 + 49 sage: L.<c> = K.absolute_field(); L Number Field in c with defining polynomial x^8 + 8*x^6 + 30*x^4 - 40*x^2 + 49
sage: from_L, to_L = L.structure() sage: from_L Isomorphism from Number Field in c with defining polynomial x^8 + 8*x^6 + 30*x^4 - 40*x^2 + 49 to Number Field in a with defining polynomial x^4 + 3 over its base field sage: from_L(c) a - b sage: to_L Isomorphism from Number Field in a with defining polynomial x^4 + 3 over its base field to Number Field in c with defining polynomial x^8 + 8*x^6 + 30*x^4 - 40*x^2 + 49 sage: to_L(a) -5/182*c^7 - 87/364*c^5 - 185/182*c^3 + 323/364*c sage: to_L(b) -5/182*c^7 - 87/364*c^5 - 185/182*c^3 - 41/364*c sage: to_L(a)^4 -3 sage: to_L(b)^2 -2
self) |
Return the chosen generator over QQ for this relative number field.
sage: y = polygen(QQ,'y') sage: k.<a> = NumberField([y^2 + 2, y^4 + 3]) sage: g = k.absolute_generator(); g a0 - a1 sage: g.minpoly() x^2 + 2*a1*x + a1^2 + 2 sage: g.absolute_minpoly() x^8 + 8*x^6 + 30*x^4 - 40*x^2 + 49
self) |
Return the polynomial over
that defines this field as an
extension of the rational numbers.
sage: k.<a, b> = NumberField([x^2 + 1, x^3 + x + 1]); k Number Field in a with defining polynomial x^2 + 1 over its base field sage: k.absolute_polynomial() x^6 + 5*x^4 - 2*x^3 + 4*x^2 + 4*x + 1
self) |
Return defining polynomial of this number field as a pair, an ntl polynomial and a denominator.
This is used mainly to implement some internal arithmetic.
sage: NumberField(x^2 + (2/3)*x - 9/17,'a').polynomial_ntl() ([-27 34 51], 51)
self) |
sage: K.<a,b> = NumberField([x^3 + 3, x^3 + 2]); K Number Field in a with defining polynomial x^3 + 3 over its base field sage: V,from_V,to_V = K.absolute_vector_space(); V Vector space of dimension 9 over Rational Field sage: from_V Isomorphism from Vector space of dimension 9 over Rational Field to Number Field in a with defining polynomial x^3 + 3 over its base field sage: to_V Isomorphism from Number Field in a with defining polynomial x^3 + 3 over its base field to Vector space of dimension 9 over Rational Field sage: c = (a+1)^5; c 7*a^2 + (-10)*a - 29 sage: to_V(c) (-29, -712/9, 19712/45, 0, -14/9, 364/45, 0, -4/9, 119/45) sage: from_V(to_V(c)) 7*a^2 + (-10)*a - 29 sage: from_V(3*to_V(b)) 3*b
self) |
Return the base field of this relative number field.
sage: k.<a> = NumberField([x^3 + x + 1]) sage: R.<z> = k[] sage: L.<b> = NumberField(z^3 + a) sage: L.base_field() Number Field in a with defining polynomial x^3 + x + 1 sage: L.base_field() is k True
This is very useful because the print representation of a relative field doesn't describe the base field.
sage: L Number Field in b with defining polynomial z^3 + a over its base field
self) |
This is exactly the same as base_field.
sage: k.<a> = NumberField([x^2 + 1, x^3 + x + 1]) sage: k.base_ring() Number Field in a1 with defining polynomial x^3 + x + 1 sage: k.base_field() Number Field in a1 with defining polynomial x^3 + x + 1
self, names) |
Return relative number field isomorphic to self but with the given generator names.
Input:
Also, K.structure()
returns from_K and to_K, where
from_K is an isomorphism from K to self and to_K is an
isomorphism from self to K.
sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K Number Field in a with defining polynomial x^4 + 3 over its base field sage: L.<c,d> = K.change_names() sage: L Number Field in c with defining polynomial x^4 + 3 over its base field sage: L.base_field() Number Field in d with defining polynomial x^2 + 2
An example with a 3-level tower:
sage: K.<a,b,c> = NumberField([x^2 + 17, x^2 + x + 1, x^3 - 2]); K Number Field in a with defining polynomial x^2 + 17 over its base field sage: L.<m,n,r> = K.change_names() sage: L Number Field in m with defining polynomial x^2 + 17 over its base field sage: L.base_field() Number Field in n with defining polynomial x^2 + x + 1 over its base field sage: L.base_field().base_field() Number Field in r with defining polynomial x^3 - 2
self, K) |
Compute all field embeddings of the relative number field self into the field K (which need not even be a number field, e.g., it could be the complex numbers). This will return an identical result when given K as input again.
If possible, the most natural embedding of K into self is put first in the list.
Input:
sage: K.<a,b> = NumberField([x^3 - 2, x^2+1]) sage: f = K.embeddings(ComplexField(58)); f [ Relative number field morphism: From: Number Field in a with defining polynomial x^3 - 2 over its base field To: Complex Field with 58 bits of precision Defn: a |--> -0.62996052494743676 - 1.0911236359717214*I b |--> -1.9428902930940239e-16 + 1.0000000000000000*I, ... Relative number field morphism: From: Number Field in a with defining polynomial x^3 - 2 over its base field To: Complex Field with 58 bits of precision Defn: a |--> 1.2599210498948731 b |--> -0.99999999999999999*I ] sage: f[0](a)^3 2.0000000000000002 - 8.6389229103644993e-16*I sage: f[0](b)^2 -1.0000000000000001 - 3.8857805861880480e-16*I sage: f[0](a+b) -0.62996052494743693 - 0.091123635971721295*I
self, [names=None]) |
Return the absolute number field
that is the Galois
closure of self.
self, [pari_group=True], [use_kash=False]) |
Return the Galois group of the Galois closure of this number
field as an abstract group. Note that even though this is an
extension
, the group will be computed as if it were
.
For more (important!) documentation, so the documentation
for Galois groups of polynomials over
, e.g., by
typing
K.polynomial().galois_group?
, where
is a number field.
sage: x = QQ['x'].0 sage: K.<a> = NumberField(x^2 + 1) sage: R.<t> = PolynomialRing(K) sage: L = K.extension(t^5-t+a, 'b') sage: L.galois_group() Galois group PARI group [240, -1, 22, "S(5)[x]2"] of degree 10 of the Number Field in b with defining polynomial t^5 + (-1)*t + a over its base field
self) |
sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K Number Field in a with defining polynomial x^4 + 3 over its base field sage: K.is_absolute() False sage: K.is_relative() True
self, [proof=None]) |
Determine whether or not
is free (i.e. if
is
a free
-module).
Input:
sage: x = QQ['x'].0 sage: K.<a> = NumberField(x^2+6) sage: L.<b> = K.extension(K['x'].gen()^2 + 3) ## extend by x^2+3 sage: L.is_free() False
self) |
Return True if this relative number field is Galois over
.
sage: k.<a> =NumberField([x^3 - 2, x^2 + x + 1]) sage: k.is_galois() True sage: k.<a> =NumberField([x^3 - 2, x^2 + 1]) sage: k.is_galois() False
self, element) |
Lift an element of this extension into the base field if possible, or raise a ValueError if it is not possible.
sage: x = QQ['x'].0 sage: K = NumberField(x^3 - 2, 'a') sage: R = K['x'] sage: L = K.extension(R.gen()^2 - K.gen(), 'b') sage: b = L.gen() sage: L.lift_to_base(b^4) a^2 sage: L.lift_to_base(b) Traceback (most recent call last): ... ValueError: The element b is not in the base field
self) |
Return the maximal order, i.e., the ring of integers of this number field.
sage: K.<a,b> = NumberField([x^2 + 1, x^2 - 3]) sage: OK = K.maximal_order(); OK.basis() [1, 1/2*a - 1/2*b, (-1/2*b)*a + 1/2, a] sage: charpoly(OK.1) x^2 + b*x + 1 sage: charpoly(OK.2) x^2 + (-1)*x + 1 sage: O2 = K.order([3*a, 2*b]) sage: O2.index_in(OK) 144
self) |
Return number of roots of unity in this relative field.
sage: K.<a, b> = NumberField( [x^2 + x + 1, x^4 + 1] ) sage: K.number_of_roots_of_unity() 24 sage: K.roots_of_unity()[:5] [(-b^3)*a, b^2*a + b^2, -b, (-1)*a, (-b^3)*a - b^3]
self) |
Return the order with given ring generators in the maximal order of this number field.
Input:
The check_is_integral and check_rank inputs must be given as explicit keyword arguments.
sage: P.<a,b,c> = QQ[2^(1/2), 2^(1/3), 3^(1/2)] sage: R = P.order([a,b,c]); R Relative Order in Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field
The base ring of an order in a relative extension is still ZZ.
sage: R.base_ring() Integer Ring
One must give enough generators to generate a ring of finite index in the maximal order:
sage: P.order([a,b]) Traceback (most recent call last): ... ValueError: the rank of the span of gens is wrong
self) |
PARI polynomial corresponding to the polynomial over the rationals that defines this field as an absolute number field.
sage: k.<a, c> = NumberField([x^2 + 3, x^2 + 1]) sage: k.pari_polynomial() x^4 + 8*x^2 + 4 sage: k.defining_polynomial () x^2 + 3
self) |
Return the PARI relative polynomial associated to this number field. This is always a polynomial in x and y.
sage: k.<i> = NumberField(x^2 + 1) sage: m.<z> = k.extension(k['w']([i,0,1])) sage: m Number Field in z with defining polynomial w^2 + i over its base field sage: m.pari_relative_polynomial () x^2 + y
self) |
Return the PARI relative number field object associated to this relative extension.
sage: k.<a> = NumberField([x^4 + 3, x^2 + 2]) sage: k.pari_rnf() [x^4 + 3, [], [[108, 0; 0, 108], [3, 0]~], ... 0]
self) |
Return the defining polynomial of this number field.
sage: y = polygen(QQ,'y') sage: k.<a> = NumberField([y^2 + y + 1, x^3 + x + 1]) sage: k.polynomial() y^2 + y + 1
This is the same as defining_polynomial:
sage: k.defining_polynomial() y^2 + y + 1
Use absolute polynomial for a polynomial that defines the absolute extension.
sage: k.absolute_polynomial() x^6 + 3*x^5 + 8*x^4 + 9*x^3 + 7*x^2 + 6*x + 3
self, [proof=None]) |
Return the relative discriminant of this extension
as
an ideal of
. If you want the (rational) discriminant of
, use e.g.
L.discriminant()
.
TODO: Note that this uses PARI's rnfdisc
function, which
according to the documentation takes an nf
parameter in
GP but a bnf
parameter in the C library. If the C
library actually accepts an nf
, then this function
should be fixed and the proof
parameter removed.
Input:
sage: K.<i> = NumberField(x^2 + 1) sage: t = K['t'].gen() sage: L.<b> = K.extension(t^4 - i) sage: L.relative_discriminant() Fractional ideal (256) sage: factor(L.discriminant()) 2^24 sage: factor( L.relative_discriminant().norm() ) 2^16
self, alpha, names) |
Given an element alpha in self, return a relative number field
isomorphic to self that is relative over the absolute field
, along with isomorphisms from
to self and
from self to K.
Input:
Also, K.structure()
returns from_K and to_K, where
from_K is an isomorphism from K to self and to_K is an isomorphism
from self to K.
sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K Number Field in a with defining polynomial x^4 + 3 over its base field sage: L.<z,w> = K.relativize(a^2) sage: z^2 z^2 sage: w^2 -3 sage: L Number Field in z with defining polynomial x^4 + (-2*w + 4)*x^2 + 4*w + 1 over its base field sage: L.base_field() Number Field in w with defining polynomial x^2 + 3
self) |
Return all the roots of unity in this relative field, primitive or not.
sage: K.<a, b> = NumberField( [x^2 + x + 1, x^4 + 1] ) sage: K.roots_of_unity()[:5] [(-b^3)*a, b^2*a + b^2, -b, (-1)*a, (-b^3)*a - b^3]
self) |
Return vector space over the base field of self and isomorphisms from the vector space to self and in the other direction.
sage: K.<a,b,c> = NumberField([x^2 + 2, x^3 + 2, x^3 + 3]); K Number Field in a with defining polynomial x^2 + 2 over its base field sage: V, from_V, to_V = K.vector_space() sage: from_V(V.0) 1 sage: to_V(K.0) (0, 1) sage: from_V(to_V(K.0)) a sage: to_V(from_V(V.0)) (1, 0) sage: to_V(from_V(V.1)) (0, 1)
The underlying vector space and maps is cached:
sage: W, from_V, to_V = K.vector_space() sage: V is W True
Special Functions: __call__,
__init__,
__reduce__,
_coerce_impl,
_fractional_ideal_class_,
_gen_relative,
_Hom_,
_latex_,
_NumberField_relative__base_inclusion,
_pari_base_bnf,
_pari_base_nf,
_repr_
self, x) |
Coerce x into this relative number field.
We construct the composite of three quadratic fields, then coerce from the quartic subfield of the relative extension:
sage: k.<a,b,c> = NumberField([x^2 + 5, x^2 + 3, x^2 + 1]) sage: m = k.base_field(); m Number Field in b with defining polynomial x^2 + 3 over its base field sage: k(m.0) b sage: k(2/3) 2/3 sage: k(m.0^4) 9
TESTS:
sage: K.<a> = NumberField(ZZ['x'].0^2 + 2, 'a') sage: L.<b> = K.extension(ZZ['x'].0 - a, 'b') sage: L(a) a sage: L(b+a) 2*a sage: K.<a> = NumberField(ZZ['x'].0^5 + 2, 'a') sage: L.<b> = K.extension(ZZ['x'].0 - a, 'b') sage: L(a) a sage: L(a**3) a^3 sage: L(a**2+b) a^2 + a sage: L.<b> = K.extension(ZZ['x'].0 + a/2, 'b') sage: L(a) a sage: L(b) -1/2*a
self) |
TESTS:
sage: Z = var('Z') sage: K.<w> = NumberField(Z^3 + Z + 1) sage: L.<z> = K.extension(Z^3 + 2) sage: L = loads(dumps(K)) sage: print L Number Field in w with defining polynomial Z^3 + Z + 1 sage: print L == K True
self, x) |
Canonical implicit coercion of x into self.
Elements of this field canonically coerce in, as does anything that coerces into the base field of this field.
sage: k.<a> = NumberField([x^5 + 2, x^7 + 3]) sage: b = k(k.base_field().gen()) sage: b = k._coerce_impl(k.base_field().gen()) sage: b^7 -3 sage: k._coerce_impl(2/3) 2/3 sage: c = a + b # this works
self) |
Return the Python class used to represent ideals of a relative number field.
sage: k.<a> = NumberField([x^5 + 2, x^7 + 3]) sage: k._fractional_ideal_class_ () <class 'sage.rings.number_field.number_field_ideal_rel.NumberFieldFractiona lIdeal_rel'>
self) |
Return root of defining polynomial, which is a generator of the relative number field over the base.
sage: k.<a> = NumberField(x^2+1); k Number Field in a with defining polynomial x^2 + 1 sage: y = polygen(k) sage: m.<b> = k.extension(y^2+3); m Number Field in b with defining polynomial x^2 + 3 over its base field sage: c = m.gen(); c b sage: c^2 + 3 0
self, codomain, [cat=None]) |
Return homset of homomorphisms from this relative number field to the codomain.
The cat option is currently ignored. The result is not cached.
This function is implicitly called by the Hom method or function.
sage: K.<a,b> = NumberField([x^3 - 2, x^2+1]) sage: K.Hom(K) Automorphism group of Number Field in a with defining polynomial x^3 - 2 over its base field sage: type(K.Hom(K)) <class 'sage.rings.number_field.morphism.RelativeNumberFieldHomset'>
self) |
Return a LaTeXrepresentation of the extension.
sage: x = QQ['x'].0 sage: K.<a> = NumberField(x^3 - 2) sage: t = K['x'].gen() sage: K.extension(t^2+t+a, 'b')._latex_() '( \\mathbf{Q}[a]/(a^{3} - 2) )[b]/(b^{2} + b + a)'
self, element) |
Given an element of the base field, give its inclusion into this extension in terms of the generator of this field.
This is called by the canonical coercion map on elements from the base field.
sage: k.<a> = NumberField([x^2 + 3, x^2 + 1]) sage: m = k.base_field(); m Number Field in a1 with defining polynomial x^2 + 1 sage: k._coerce_(m.0 + 2/3) a1 + 2/3 sage: s = k._coerce_(m.0); s a1 sage: s^2 -1
This implicitly tests this coercion map:
sage: K.<a> = NumberField([x^2 + p for p in [5,3,2]]) sage: K._coerce_(K.base_field().0) a1 sage: K._coerce_(K.base_field().0)^2 -3
self, [proof=None]) |
Return the PARI bnf (big number field) representation of the base field.
Input:
sage: k.<a> = NumberField([x^3 + 2, x^2 + 2]) sage: k._pari_base_bnf() [[;], matrix(0,9), [;], ... 0]
self) |
Return the PARI number field representation of the base field.
sage: y = polygen(QQ,'y') sage: k.<a> = NumberField([y^3 + 2, y^2 + 2]) sage: k._pari_base_nf() [y^2 + 2, [0, 1], -8, 1, ..., [1, 0, 0, -2; 0, 1, 1, 0]]
self) |
Return string representation of this relative number field.
The base field is not part of the string representation. To
find out what the base field is use self.base_field()
.
sage: k.<a, b> = NumberField([x^5 + 2, x^7 + 3]) sage: k Number Field in a with defining polynomial x^5 + 2 over its base field sage: k.base_field() Number Field in b with defining polynomial x^7 + 3
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