Show that a $ \ 2$-adic integer not divisible by $4$ is a square

Question

Show that a $ \ 2$-adic integer not divisible by $4$ is a square iff it is congruent to $1$ $\pmod 8 $.

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My try:

We know that $a \equiv 1 \pmod 8$ implies $a$ is a square modulo $2^n$ for all $n$.

Hence the converse part can be proved using this but how to prove the first implication?

Answer 1

If a $2$-adic integer $x$ is an odd square, say $x=y^2$, then $x\equiv y^2\pmod{8}$ and hence...