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 field F together with an automorphism, or, equivalently, the group-completion of the E∞-algebra
of maps from S$^1$ to the disjoint union of classifying spaces BGL$_d$(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. We also discuss the topological
cases of F = C, R