Segal K-theory of vector spaces with an automorphism

We describe the Segal K-theory of the symmetric monoidal category of finite-dimensional vector spaces over a perfect field F together with an automorphism, or, equivalently, the group-completion of the E_-algebra of maps from S¹ to the disjoint union of classifying spaces BGLd (F), in terms of the K-theory of finite field extensions of F. A key ingredient for this is a computation of the Segal K-theory of the category of finite-dimensional vector spaces with a nilpotent endomorphism, which we do over any field F. We also discuss the topological cases of F = C, R.