# Absolute values in $K(x)$ are non-archimedean.

I was reading this article http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/ostrowskiF(T).pdf in which they prove the Ostrowsky theorem for $K(x)$ where K is a field.

However I noticed that they assume that the absolute value in $K(x)$ is non-archimedean and I don't really find an easy way to prove this.

• They assume that the valuation is trivial on $K$. Isn't it a fact that if a valuation is bounded on the prime field then it is non-archimedean? – Jyrki Lahtonen Dec 3 '17 at 21:28

On the first page of the paper you link to, "example 1" actually shows that this is not true in general. E.g. on $\mathbb{Q}(x)$ one can define lots of archimedean valuations. Namely, choose any number $\alpha \in \mathbb{C}$ which is transcendent (over $\mathbb{Q}$), and check that sending $x$ to $\alpha$ defines an isomorphism $\mathbb{Q}(x) \simeq \mathbb{Q}(\alpha) \subset \mathbb{C}$. Then restrict the usual absolute value of $\mathbb{C}$ to that subfield and pull it back to $\mathbb{Q}(x)$. Compare also the answer ("Oh yes there are!") to the related (but ill-posed) question: There are no archimedean function fields.
That is why after the examples, that paper explicitly works with the extra assumption that the restriction of the valuation to $K$ is trivial, meaning that $|x| = 1$ for all $x \in K^*$. Under this assumption, the valuation on $K(x)$ is indeed automatically non-archimedean. Depending on what your definition of "non-archimedean" is, this is more or less trivial, as it means that for all $x\in K$,
$|nx| = 1$ for all $n\in\mathbb{N}$.