# Extension of 2-adic valuation to the real numbers

I just want to know what properties of valuations extend to $$\mathbb R$$...

Denote an extension of the 2-adic valuation from $$\mathbb Q$$ to $$\mathbb R$$ by $$\nu$$. Suppose $$\nu(x)=\nu(y)=0$$.

Is it true that $$\nu(x+y)\ne 0$$?

What about $$\nu(x^2+y^2)\le 1$$?

I'm interested in knowing both whether these are true for every extension, as well as knowing whether there is some extension for which they are true (for every $$x$$ and $$y$$).

• I changed the title to something more appropriate, so no more clickbait. – KConrad Apr 26 at 15:23
• I saw nothing wrong with the title actually. Nothing wrong with a bit of humour. – RP_ Apr 26 at 16:03

No. The important thing to know is that, if $$K \subseteq L$$ is a field extension and $$v: K \to \mathbb{R}$$ is a valuation, then $$v$$ can be extended to $$L$$. So I can answer all of your questions by working in some easy to handle subfield of $$\mathbb{R}$$. I'll work in $$K = \mathbb{Q}(\sqrt{5})$$ for the first question and in $$K = \mathbb{Q}(\sqrt{3})$$ for the second.
The ring of integers in $$\mathbb{Q}[\sqrt{5}]$$ is $$\mathbb{Z}[\tau]$$ where $$\tau = \tfrac{1+\sqrt{5}}{2}$$, with minimal polynomial $$\tau^2=\tau+1$$. Note that $$\mathcal{O}_K/(2 \mathcal{O}_K)$$ is the field $$\mathbb{F}_4$$ with four elements. Your first statement is true in $$\mathbb{Q}$$ only because $$\mathbb{Z}/(2 \mathbb{Z})$$ has two elements.
Specifically, both $$1$$ and $$\tau$$ are in $$\mathcal{O}_K$$ but not $$2 \mathcal{O}_K$$, so $$v(1) = v(\tau) = 0$$, but $$1+\tau$$ is also not in $$2 \mathcal{O}_K$$ so $$v(1+\tau)=0$$ as well.
Similarly, the ring of integers in $$\mathbb{Q}(\sqrt{3})$$ is $$\mathbb{Z}[\sqrt{3}]$$ and the prime $$2$$ is ramified, with $$2 = (1+\sqrt{3})^2 (2-\sqrt{3})$$ (note that $$2-\sqrt{3}$$ is a unit). We have $$v(1) = v(\sqrt{3}) = 0$$, but $$v(1+\sqrt{3}^2) = 2$$. In this case, the result is true in $$\mathbb{Q}$$ because $$2$$ is unramified.
• "the field $\Bbb F_4$ with four elements"? – Greg Martin Apr 26 at 16:47