I am trying to understand the definition of $p$-adic integers. I know there are several posts on MSE about that but I still couldn't understand. I am looking at the power series definition:
The $p$-adic integers are the set of formal power series
$$
a_0+a_1p+a_2p^2+\dotsb +a_np^n+\dotsb
$$
where $p$ is a prime and $a_k\in \{0,1,\dotsc,p-1\}$
Example: Let $p=2$. Any natural number can be expressed in the aforementioned formal power series form. I don't see any other number other than the natural numbers that can be expressed in the above formal power series form. Is that correct?
Also, as a set it is said that the set of formal power series is bijective with $\mathbb{Z}_p$. May I know what is the explicit bijection?
Best Answer
Every rational number with an odd denominator can be expressed as such a power series. For example, $-1$ is the one in which all coefficients equal $1$, and $\frac13$ is the one with $a_0$ and all odd-index coefficients equal to $1$, and all even-index coefficients from $a_2$ on equal to $0$. Writing them in a standard notation for $p$-adics, we have: $$\ldots11111.$$ for $-1$, and $$\ldots01011.$$ for $\frac13$.
You can verify that adding $1$ to the first number produces $0$, and that multiplying the second one by $3$ produces $1$.
If you're worried about those "carry" digits trailing off to the left, remember the $p$-adic absolute value: Out to the left is where the number gets "small" anyway, so we get convergence in cases where the usual absolute value would not work for us.
The bijection you ask about is simply that the power series with coefficients $a_0, a_1, a_2$, etc. corresponds to the $p$-adic number whose digits are those coefficients: $$\ldots a_2 a_1 a_0.$$