I am currently trying to understand the basics of $p$-adic integers using http://mathworld.wolfram.com/p-adicNumber.html.
I quote "the equation $x^2=2$ can easily be shown to have no solutions in the field of $2$-adic numbers (we simply take the valuation of both sides)". I tried to prove it to see whether I handle the definitions properly. My proof is as follows:
Assume there exists a rational number $x = \frac{r}{s}\cdot 2^a$ for which $x^2 = 2$, where $r,s$ are odd and $a \in \mathbb{Z}$. Then, $x^2 = \frac{r^2}{s^2}\cdot2^{2a}$, hence $|x^2|_2 = \frac{1}{2^{2a}}$ for $a \in \mathbb{Z}$. However, $|2|_2 = \frac{1}{2}$. It follows that $1 = 2a$ which cannot be true because both sides have opposing parity.
Am I doing this correct?