TITLE: Representations of p-adic integers as certain infinite sums
QUESTION [6 upvotes]: One way to define the p-adic integers is as the $p$-adic completion of $\mathbb{Z}$. With some additional work, it can be shown that this is isomorphic to $\mathbb{Z}[[x]]/(x-p)$.
Now, I know that another approach is to define $\mathbb{Z}_p$ as the ring of power series with powers of $p$ and coefficients from ${0, 1,..., p-1}$.
My question is: How can we see that the first definition of $\mathbb{Z}_p$ coincides with the second definition? and how can we find an explicit isomorphism from $\mathbb{Z}[[x]]/(x-p)$ to the ring of power series described above?

REPLY [5 votes]: The problem of showing that $\mathbb Z[[x]]/(x-p)$ coincides with the $p$-adic completion of $\mathbb Z$ has been addressed a couple of times already on this site: see here and here.
Recall that the $p$-adic completion of $\mathbb Z$ is the projective limit
$\mathbb Z_p := \varprojlim_n \mathbb Z/p^n$.
The isomorphism of $\mathbb Z[[x]]/(p-x)$ with $\mathbb Z_p$ is obtained as the projective limit of the maps
$\mathbb Z[[x]]/(x-p) \to \mathbb Z[[x]]/(x-p,x^n) = \mathbb Z/p^n$, the first
map being the natural surjection, and the second map being obtained by setting $x = p$.
The isomorphism of $\mathbb Z_p$ with the "power series" that you describe is
pretty straightforward: an element of $\mathbb Z/p^n$ has a unique representative in the form $a_0 + a_1 p + \cdots a_{n-1} p^{n-1}$ with
$0 \leq a_i < p$.  An element of $\mathbb Z_p$ thus can be described in a unique
way as a "power series" $\sum_{i \geq 0} a_i p^i$ with $0 \leq a_i < p$.
Composing the two indicated isomorphisms gives the isomorphism you asked about.
It is pretty explicit: given an element of $\mathbb Z[[x]]/(x-p)$, represented
by some power series $f(x)$, to get the first $n$ terms of the associated "power series", I take the degree $\leq n-1$ part of $f(x)$, replaced $x$ by $p$,
reduce modulo $p^n$, and then choose the coset representative of the form
$a_0 + \cdots a_{n-1} p^{n-1}$ as above.