The Galois-equivariant $K$-theory of finite fields

We compute the $RO(G)$-graded equivariant algebraic $K$-groups of a finite field with an action by its Galois group $G$. Specifically, we show these $K$-groups split as the sum of an explicitly computable term and the well-studied $RO(G)$-graded coefficient groups of the equivariant Eilenberg–MacLane spectrum $H\underset{‾}{\mathbb{Z}}$. Our comparison between the equivariant $K$-theory spectrum and $H\underset{‾}{\mathbb{Z}}$ further shows they share the same Tate spectra and geometric fixed point spectra. In the case where $G$ has prime order, we provide an explicit presentation of the equivariant $K$-groups.