Why is $0.1 =1$ in the $p$-adic numbers?

Question

I have been looking at $p$-adic numbers recently. Unless I am mistaken, if $$ \alpha = \sum_{i=0}^\infty a_ip^i $$ then we write $$ \alpha = 0.a_0a_1a_2a_3\dots $$ With this we would get that $1 = 0.1$. That is, the multiplicative identity in the field $\mathbb{Q}_p$ is $0.1$.

What I don't understand is why we don't say that $$ \alpha = \sum_{i=0}^\infty a_ip^i $$ is equal to $a_0.a_1a_2a_3\dots$

instead so that $1=1$.

EDIT: I would like an answer so that I can accept. If you would include in that answer a discussion of why the convention above is good for $p$-adic numbers, that would be appreciated.

Answer 1

I think the most compelling reasons are that

1. It is common to solve algebraic equations over the integers by proving a priori bounds on their size, finding all solutions over $\mathbb{F}_p$ for some "good" prime $p$, Hensel lifting the solutions digit by digit until we reach the bounds and either find or rule out an integer solution. Prominent examples include

   • factorization of polynomials over the integers using the Zassenhaus–Berlekamp method and
   • producing points on hyperelliptic cuves using the Chabauty–Coleman method.

2. We frequently use continuity of $p$-adic functions to summarize infinite families of congruence relations among integers. For example,

   • $p$-adic continuity of the binomial coefficients encompasses Kummer's theorem, Lucas's theorem, and Fermat's little theroem,
   • $p$-adic continuity of the $p$-adic gamma function entails Wilson's theorem, and
   • $p$-adic continuity of the $p$-adic zeta function is equivalent to Kummer's congruences.