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 Z. Our comparison between the equivariant K-theory spectrum and H 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.