TITLE: Extending 2-adic valuation to real numbers
QUESTION [10 upvotes]: When proving Monsky's theorem, one of the steps, which, from what I have so far seen, no proof can avoid, is extending the 2-adic valuation to all real numbers, so that it still satisfies $|xy|_2=|x|_2|y|_2$ and $|x+y|_2\leq\max\{|x|_2,|y|_2\}$. However, no proof of Monsky's theorem which I have seen explained why this extension exists. One of the proofs has mentioned that the proof of existence requires axiom of choice. This made me think that standard argument using Zorn's lemma would do the trick, but I have realized that giving arbitrary value to an irrational will not always lead to consistent valuation (e.g. we can't set $|\sqrt{2}|_2=1$, as this would lead to $|2|_2=|\sqrt{2}\cdot\sqrt{2}|_2=|\sqrt{2}|_2^2=1\neq\frac{1}{2}=|2|_2$).

How does one prove existence of an extension of $|\cdot|_2$ as specified above?

REPLY [4 votes]: How does one prove the existence of an extension of $|\cdot|_2$ as specified above?

Any p-adic valuation on $\mathbb{Q}$ can be extended to $\mathbb{R}$ or $\mathbb{C}$. The construction is as follows: since there exists a field isomorphism $T:\mathbb{C}\to\mathbb{C}_p$ that extends the identity map in $\mathbb{Q}$, you can define the valuation $|\cdot|_*:\mathbb{C}\to\mathbb{R}$ by $|x|_*=|T(x)|_p$. It is not hard to see that $|x+y|_*\leq\max\{|x|_*,|y|_*\}$ for all $x,y\in\mathbb{C}$. Then, the restriction of $|\cdot|_*$ to $\mathbb{R}$ is a valuation on $\mathbb{R}$ that extends the $p$-adic valuation of $\mathbb{Q}$. Of course, this construction cannot be done explicitly since there is not an explicit formula for $T$. We only know about its existence. A proof of the fact that $\mathbb{C}$ and $\mathbb{C}_p$ are isomorphic can be found on page 83 of the book: Non-Archimedean Functional Analysis - [A.C.M. van Rooij] - 1978.
More generally, it has been proven (see here) that any two algebraically closed fields of the same characteristic and cardinality are isomorphic.