I was recently introduced to the idea of 2-adic norms for numbers in $\mathbb{Q}$. I was told that the rules can be expanded into $\mathbb{R}$, but I am unclear of how to evaluate them. For example, what would $|12|_{2}$ evaluate to? or $|\sqrt{2}|_{2}$
This is to help me understand (in some small way) Monsky's Theorem for my combinatorics class.
The result is technical and non-constructive - you can't actually define an extension of the $p$-adic norm to all real numbers, but you can prove (at least if you assume the Axiom of Choice) that there exists such an extension. And if there is an extension, then there are a huge number of different extensions.
So there isn't a natural value for $|\pi|_p$.
However, given an algebraic number, $\alpha$, there is a well-defined value for $|\alpha|_p$. Specifically, if $\alpha$ is the root of the minimal rational polynomial:
$$x^n+a_{n-1}x^{n-1}+\cdots a_1x+a_0$$
Then there is a formula for $|\alpha|_p$ in terms of tha $a_i$.
I think that formula is $|a_0|^{1/n}$, but I could be wrong about that.
For algebraic numbers of the form $\sqrt[k]{q}$ it is definitely the case that $|\sqrt[k]q|_p=|q|_p^{1/k}$.