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.
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
p-adic behavior
of the function
qn−1.
We use several 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-algebra
THH∗(K(Fq);HFp), and we
compute
V(1)∗THH(K(Fq))
in the first two cases.
