  We describe the Segal $K$-theory of the symmetric monoidal category of
finite-dimensional vector spaces over a perfect field $\mathbb{F}$ together
with an automorphism, or, equivalently, the group-completion of the
$E_\infty$-algebra of maps from $S^1$ to the disjoint union of classifying
spaces $\mathrm{BGL}_d(\mathbb F)$, in terms of the $K$-theory of finite field
extensions of $\mathbb{F}$. 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 $\mathbb F$. We also discuss
the topological cases of $\mathbb F =\mathbb C,\mathbb R$.
