How do you construct a quotient ring in Sage?
We create the quotient ring
, and demonstrate many
basic functions with it.
sage: R = PolynomialRing(GF(97),'x') sage: x = R.gen() sage: S = R.quotient(x^3 + 7, 'a') sage: a = S.gen() sage: S Univariate Quotient Polynomial Ring in a over Finite Field of size 97 with modulus x^3 + 7 sage: S.is_field() True sage: a in S True sage: x in S True sage: S.polynomial_ring() Univariate Polynomial Ring in x over Finite Field of size 97 sage: S.modulus() x^3 + 7 sage: S.degree() 3
In Sage, in
means that there is a ``canonical coercion'' into
the ring. So the integer
and
are both in
,
although
really needs to be coerced.
You can also compute in quotient rings without actually computing
then using the command quo_rem
as follows.
sage: R = PolynomialRing(GF(97),'x') sage: x = R.gen() sage: f = x^7+1 sage: (f^3).quo_rem(x^7-1) (x^14 + 4*x^7 + 7, 8)
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