I was reading the book P-adic numbers: An introduction by Fernando Gouvea and I found the following problem:
Let $m \in \mathbb{Z}$, and suppose that the congruence $X^2 \equiv m\pmod p$ has a solution; show that if $p \neq 2$ and $p \nmid m$ it is always possible to "extend" this solution to a full coherent sequence of solutions of $X^2 \equiv m\pmod {p^n}$. Use this to find a necessary and sufficient condition for the equation $X^2 = m$ to have a root in $\mathbb{Q}_p$ for $p \neq 2$. What is especial about $p = 2$?
At this point in the book, we do not know Hensel's lemma. How can we infer a necessary and sufficient condition for the equation $X^2 = m$ to have a root in $\mathbb{Q}_p$ for $p \neq 2$ from "scratch" (not using Hensel's lemma)? Thanks in advance for your help!