Valuation theory

Question

what is the the natural, non-trivial valuation $\mathbb{v}$ in a field F? This term appeared in some articles about $\mathbb{R}$-places that I was studying. I do have some small background in valuation theory, but I don't know what "natural valuation" stands for.

Answer 1

There is not "the" natural non-trivial (absolute value) valuation on a field $F$. A finite field only has the trivial valuation, for example, given by $|a|=1$ for all $a\neq 0$. The field $\Bbb Q$ has a $p$-adic valuation $|\cdot |_{p}$ for every prime $p$ and also the usual absolute valuation $|\cdot |_{\infty}$. The first is non-Archimedean, the second one is Archimedian. The Theorem of Ostrowski says that every non-trivial valuation on $\Bbb Q$ is equivalent to either $|\cdot |_{p}$ or $|\cdot |_{\infty}$.