The topological Hochschild homology of algebraic K-theory of finite fields

Let K(Fq) be the algebraic K-theory spectrum of the finite field with q elements and let p ≥ 5 be a prime number coprime to q. In this paper we study the mod p and v1 topological Hochschild homology of K(Fq), denoted V (1)∗ THH(K(Fq)), as an Fp-algebra. The computations are organized in four different cases, depending on the mod p behaviour of the function q−1. We use different spectral sequences, in particular the Bökstedt spectral sequence and a generalization of a spectral sequence of Brun developed in an earlier paper. We calculate the Fp-algebras THH∗(K(Fq);HFp), and we compute V (1)∗ THH(K(Fq)) in the first two cases.