Step-by-step method for $p$-adic expansion of $\sqrt{a}$

Question

Let $a>0$.

I need to check following:

1. Is $\sqrt{a}$ an element of $\mathbb{Q}_p$?
2. How I can find the $p$-adic expansion for $\pm\sqrt{a}$?
3. How I can distinguish $\sqrt{a}$ and $-\sqrt{a}$?

Example: $5$-adic expansion for $\sqrt{6},\sqrt{7}$.

I am a beginner and do not know much about specific calculations.

Answer 1

First you should check if there even is a square root mod $p$.

Suppose $(n,p) = 1$ and $n$ is a square mod $p$, so the square root in ${\mathbb Q}_p$ should exist. One quick way to explicitly find good approximations of it is Newton's method. The iteration is $x_{k+1} = \frac{x_k + n/x_k}{2}$ where $x_0$ is an integer with $n \equiv x_0^2 \mod p$. Then $x_{k+1}^2 - n = (x_k - n)^2/4$ so $\|x_{k+1}^2 - n\|_p = \|x_k^2 - n\|_p^2$. Since $\|x_0^2 - n\|_p \le p^{-1}$ we get $\|x_k^2 - n\|_p \le p^{-2^k}$.