I'm trying to find a name for the following quantity that came up in my research. I've asked some people and looked around myself but can't find a name, yet it seems like something that has probably been studied before.
Let $K$ be a field and $f(x)$ a polynomial in $K[x]$, and further assume that $\bar K$ has a norm $|\cdot |$. Let $\alpha,\beta$ be the roots of smallest and largest norm, respectively. What would one call, and are there any references you know of related to: $$\frac{\log|\alpha|}{\log|\beta|}?$$ (or its reciprocal, of course)
I'm mainly interested in the case where $K$ is a local field, so really $\log|\alpha|$ is the valuation of $\alpha$ - for this reason I prefer not to assume that $f(x)$ is irreducible over $K$.
