The $K$-theory of finite Tambara fields: away from $p$

TL;DR

This paper advances the understanding of algebraic K-theory for finite Tambara fields at odd primes, showing torsion properties and patterns predicted by prior conjectures, with computational evidence for p-torsion.

Contribution

It introduces a new approach to compute K-theory of constant C_{p^n}-Tambara fields at odd primes, establishing torsion properties and explicit patterns after inverting p.

Findings

K-theory groups are torsion for these fields.

The groups follow a simple pattern away from p.

p-power torsion is generally nontrivial, confirmed by computer calculations.

Abstract

In previous work, the author and Chan computed the algebraic $K$-theory of the constant $C_2$-Tambara field with value the field with two elements, using a method which fails at odd primes. Herein we make progress towards the corresponding odd primary computations using a completely new idea. Particularly, we show that the $K$-theory groups of any constant $C_{p^n}$-Tambara field with value a characteristic $p$ finite field are torsion, and we completely determine these groups after inverting $p$. The away-from-$p$-torsion satisfies a simple pattern predicted by previous work, and a computer-aided computation shows that the $p$-power torsion is nontrivial in general.