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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.

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    $\begingroup$ $f \mapsto q^{-\deg(f)}$ 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 $\mathbb{C}$. $\endgroup$ – reuns Jan 8 at 0:54

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