Let $K$ be a field, and $K(x)$ the field of rational functions over $K$. Consider the degree valuation $v$ on $K(x)$, That is
$v\left(\frac{f(x)}{g(x)}\right)=\deg(g)-\deg(f)$. So for every $f(x)\in K[x]$, $v(f(x))\leq 0$.

Now let $F$ be a finite extension of $K(x)$, and $R$ the integral closure of $K[x]$ in $F$. Let $w$ be some extension of $v$ to a valuation on $F$. My question is, if for every $a\in R$, it happens that $w(a)\leq 0$.

I checked it for some private cases, like $F=K(x^{1/n})$, and $w$ some extension of $v$, and it turned out to be true. But I can't manage to prove it for the general case.

P.S. I also asked this question on Math.SE, but I have a feeling it won't get an answer there.

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