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

## TL;DR

This paper computes the topological Hochschild homology of algebraic K-theory spectra of finite fields at primes p ≥ 5, using spectral sequences to analyze different cases based on the behavior of q^n-1.

## Contribution

It provides explicit calculations of topological Hochschild homology for algebraic K-theory of finite fields, extending previous spectral sequence methods to new cases.

## Key findings

- Computed $THH_*(K( ext{finite field})); H ext{F}_p$ explicitly.
- Determined $V(1)_*THH(K( ext{finite field}))$ in two cases.
- Organized computations based on the mod p behavior of $q^n-1$.

## Abstract

Let $K(\mathbb{F}_q)$ be the algebraic $K$-theory spectrum of the finite field with $q$ elements and let $p \geq 5$ be a prime number coprime to $q$. In this paper we study the mod $p$ and $v_1$ topological Hochschild homology of $K(\mathbb{F}_q)$, denoted $V(1)_*THH(K(\mathbb{F}_q))$, as an $\mathbb{F}_p$-algebra. The computations are organized in four different cases, depending on the mod $p$ behaviour of the function $q^n-1$. We use different spectral sequences, in particular the B\"okstedt spectral sequence and a generalization of a spectral sequence of Brun developed in an earlier paper. We calculate the $\mathbb{F}_p$-algebras $THH_*(K(\mathbb{F}_q); H\mathbb{F}_p)$, and we compute $V(1)_*THH(K(\mathbb{F}_q))$ in the first two cases.