Can someone please explain what the characteristic polynomial is in the case of an element over a field in the case below from Serre's Local Fields. I have only ever seen this phrase with matrices and not in this context:
Proposition: Let R be a discrete valuation ring, K its field of fractions, and let L be an extension of K of finite degree n. Let B be the integral closure of R in L. Suppose that B is a discrete valuation ring and that the residue field $\bar{L}$ of B is a simple extension of degree n of the residue field k=$\bar{K}$ of R. Let x be any element of B whose image $\bar{x}$ in $\bar{L}$ generates $\bar{L}$ over k, and let f be the characteristic polynomial of x over K.
Multiplication by $x$ is a linear map of the $n$-dimensional $K$-vector space into itself.