25.2 Terminology and types of $ p$ -adics

To write down a general $ p$ -adic element completely would require an infinite amount of data. Since computers do not have infinite storage space, we must instead store finite approximations to elements. Thus, just as in the case of floating point numbers for representing reals, we have to store an element to a finite precision level. The different ways of doing this account for the different types of $ p$ -adics.

We can think of $ p$ -adics in two ways. First, as a projective limit of finite groups:

$\displaystyle \mathbf{Z}_p= \lim_{\leftarrow n} \mathbf{Z} / p^n \mathbf{Z}.$

Secondly, as Cauchy sequences of rationals (or integers, in the case of $ \mathbf{Z}_p$ ) under the $ p$ -adic metric. Since we only need to consider these sequences up to equivalence, this second way of thinking of the $ p$ -adics is the same as considering power series in $ p$ with integral coefficients in the range 0 to $ p-1$ . If we only allow nonnegative powers of $ p$ then these power series converge to elements of $ \mathbf{Z}_p$ , and if we allow bounded negative powers of $ p$ then we get $ \mathbf{Q}_p$ .

Both of these representations give a natural way of thinking about finite approximations to a $ p$ -adic element. In the first representation, we can just stop at some point in the projective limit, giving an element of $ \mathbf{Z} / p^n \mathbf{Z}$ . As $ \mathbf{Z}_p/ p^n\mathbf{Z}_p\cong \mathbf{Z} / p^n \mathbf{Z}$ , this is is equivalent to specifying our element modulo $ p^n\mathbf{Z}_p$ .

The absolute precision of a finite approximation $ \bar{x} \in \mathbf{Z} / p^n \mathbf{Z}$ to $ x \in \mathbf{Z}_p$ is the non-negative integer $ n$ .

In the second representation, we can achieve the same thing by truncating a series

$\displaystyle a_0 + a_1 p + a_2 p^2 + \cdots
$

at $ p^n$ , yielding

$\displaystyle a_0 + a_1 p + \cdots + a_{n-1} p^{n-1} + O(p^n).
$

As above, we call this $ n$ the absolute precision of our element.

Given any $ x \in \mathbf{Q}_p$ with $ x \ne 0$ , we can write $ x = p^v u$ where $ v \in \mathbf{Z}$ and $ u \in \mathbf{Z}_p^{\times}$ . We could thus also store an element of $ \mathbf{Q}_p$ (or $ \mathbf{Z}_p$ ) by storing $ v$ and a finite approximation of $ u$ . This motivates the following definition: The relative precision of an approximation to $ x$ is defined as the absolute precision of the approximation minus the valuation of $ x$ . For example, if $ x = a_k p^k + a_{k+1} p^{k+1} +
\cdots + a_{n-1} p^{n-1} + O(p^n)$ then the absolute precision of $ x$ is $ n$ , the valuation of $ x$ is $ k$ and the relative precision of $ x$ is $ n-k$ .

There are four different representations of $ \mathbf{Z}_p$ in SAGE and two representations of $ \mathbf{Q}_p$ :



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