# Norm on k[X] or Q[x]?

While considering specific examples of norms on number fields, I was considering $$\mathbb{Q}[\sqrt{a}]\cong \frac{\mathbb{Q}[x]}{(x^2-a)}$$. This led me to the following question:

Given a field $$k$$, let $$k[x]$$ be the polynomial ring over $$k$$. Then the degree of $$f\in k[x]$$ gives one a norm on $$k[x]$$. Does there exist another norm on $$k[x]$$ irrespective of the base field $$k$$?

I'm not sure how to find this even for $$\mathbb{Q}[x]$$. I would appreciate any hints/references.

• $f \mapsto q^{-\deg(f)}$ is a non Archimedian absolute value on $k[x]$. For any number field $k$ let $\sigma : k \to \mathbb{C}$ and $f^\sigma(x)= \sum_{n=0}^{\deg(f)} \sigma(a_n) x^n$ then $f \mapsto |f^\sigma(\pi)|$ is an Archimedian absolute value on $k[x]$. Of course $\pi$ can be replaced by any transcendental complex number. If $q >1$ then the completion of $k(x), f \mapsto q^{-\deg(f)}$ is $k((x))$. The completion of $k(x), f \mapsto |f^\sigma(\pi)|$ is $\cong \mathbb{R}$ or $\mathbb{C}$. – reuns Jan 8 at 0:54