Let
$K\left({\mathbb{\mathbb{F}}}_{q}\right)$ be the algebraic
$K$theory spectrum of the
finite field with
$q$ elements
and let
$p\ge 5$ be a prime
number coprime to
$q$.
We study the mod
$p$
and
${v}_{1}$ topological
Hochschild homology of
$K\left({\mathbb{\mathbb{F}}}_{q}\right)$,
denoted
$V{\left(1\right)}_{\ast}THH\left(K\left({\mathbb{\mathbb{F}}}_{q}\right)\right)$, as
an
${\mathbb{\mathbb{F}}}_{p}$algebra.
The computations are organized in four different cases, depending on the
$p$adic behavior
of the function
${q}^{n}1$.
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
${\mathbb{\mathbb{F}}}_{p}$algebra
${THH}_{\ast}\left(K\left({\mathbb{\mathbb{F}}}_{q}\right);H{\mathbb{\mathbb{F}}}_{p}\right)$, and we
compute
$V{\left(1\right)}_{\ast}THH\left(K\left({\mathbb{\mathbb{F}}}_{q}\right)\right)$
in the first two cases.
