Ring of p-adic integers and harmonic sums

Question

I am trying to understand p-adic integers $\mathbb{Z}_p$ and want to ask two questions:

1. For a fixed rational number $\frac{a}{b}$ and a prime $p$, I can write $p$-adic expansion of the fraction after some calculations, most of the time.(For example $\cfrac{2}{5} \in \mathbb{Z}_3$ since it is equal to $1+3+2.3^2+3^3+\dots \in \mathbb{Z}_3$ ). However, I do not know a general algorithm. So, my question is, for any odd number $n$, how can I show that $\cfrac{1}{n} \in \mathbb{Z}_2?$
2. What is the structure of $p^k \mathbb{Z}_p$ for some positive integer $k$? Is it the ring of power series $\displaystyle p^k \sum_{n=0}^{\infty} a_n p^n = \sum_{n=k}^{\infty} a_n p^n?$

Answer 1

1. To show a number exists as a p-adic integer, we can write it as an equation and check Hensel's lemma. Hensel's lemma is essentially a canned induction argument waiting for you to confirm the base case and that a necessary step for lifting is satisfied. To do this we have to check that:

$$f(x) \equiv 0 \mod p$$ $$f'(x) \not \equiv 0 \mod p$$

So, for your choice you want to show $x=\frac{a}{b}$ is in $\mathbb{Z}_p$, then all we need to do is rearrange it into $$f(x) = bx-a = 0$$ and check the two above conditions.

There is more to say, since just failing to satisfy those conditions doesn't necessarily mean it's not a p-adic integer. It's probably best to check out the proof for Hensel's lemma to get a better understanding since that itself shows from those two starting conditions how to construct the base p expansion of a p-adic integer.

2. It's nearly the ring of formal power series but there's an important difference, we keep the $a_i$ to be from a distinct residue set; a common choice is $\{0,1,...,p-1\}$. When we add the two p-adic numbers we are forced to "carry" which doesn't happen if we treat the powers $p^n$ as distinct. This is not the only choice either, there are the Teichmuller characters which satisfy $\omega^{p-1} = 1$ that can also be used as "digits" of a p-adic number. To add and multiply them in this form the analogue for "carrying" is a different procedure and is described by Witt vectors.