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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!

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  • $\begingroup$ You want to take a solution modulo $p$, say $X_1\in\Bbb Z$, and improve it by adding a multiple of $p$, so that $X_2=X_1+\delta_1p$ now satisfies the condition $X_2^2\equiv m\pmod {p^2}$. Just do it. $\endgroup$ – Lubin May 4 '17 at 18:42
  • $\begingroup$ I did it, and it worked! Thank you for your help @Lubin. However, what about the second question? What is especial about the case $p = 2$? I tried to do the same, but it did not work. $\endgroup$ – Claudia Prune May 4 '17 at 18:59
  • $\begingroup$ I already proved it: $m$ has a root in $\mathbb{Q}_2$ if and only if $x^2 \equiv m(\mod 8)$ $\endgroup$ – Claudia Prune May 4 '17 at 19:11
  • $\begingroup$ Good. I think you’ll like Gouvêa’s book — I think it’s the best on the subject, at that level. $\endgroup$ – Lubin May 5 '17 at 2:34

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