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Let $f$ be a separable irreducible polynomial of degree $n$ with $f \in F[x]$, where $F$ is a function field. I try to understand the following part of an article:

..., we use the minimum infinite valuation of the roots $\alpha_i$ of $f$. When $F$ is a rational function field, the infinite valuation is the negative of the degree of a root. We compute these valuation at all infinite places of F and call the smallest one $\mathit{v}_0$.

Hence $\mathit{v}_\infty(\alpha_i) \geq {\mathit{v}}_{0}$ and $\mathit{v}_0 \leq 0$.

As it is written, if $F$ is a rational function field the infinite valuation is the negative of the degree. Additionally there is onyl one place at infinity, let's say $\mathfrak{p}_\infty$, so $\mathit{v}_0 = \mathfrak{p}_\infty$. However I am confused by the meaning of that.

What is the meaning of $\mathfrak{p}_\infty \leq 0$ and why should $\mathit{v}_\infty(\alpha_i) = -deg(\alpha_i) \geq \mathfrak{p}_\infty$?

Is my understanding of $\mathit{v}_0$ wrong? Right now the two inequalities look like a contradiction for me.

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