A classification of $C_{p^n}$-Tambara fields

TL;DR

This paper classifies $C_{p^n}$-Tambara fields, showing they are coinduced from simpler field-like Tambara functors and analyzing their structure based on characteristic and Frobenius behavior.

Contribution

It provides a comprehensive classification of $C_{p^n}$-Tambara fields, detailing their structure and behavior depending on characteristic and Galois extension properties.

Findings

$C_{p^n}$-Tambara fields are coinduced from $C_{p^s}$-Tambara functors.

If characteristic ≠ p, the Tambara functor is fixed-point.

If characteristic = p, all forms are characterized via Frobenius and trace analysis.

Abstract

Tambara functors arise in equivariant homotopy theory as the structure adherent to the homotopy groups of a coherently commutative equivariant ring spectrum. We show that if $k$ is a field-like $C_{p^n}$-Tambara functor, then $k$ is the coinduction of a field-like $C_{p^s}$-Tambara functor $\ell$ such that $\ell(C_{p^s}/e)$ is a field. If this field has characteristic other than $p$, we observe that $\ell$ must be a fixed-point Tambara functor, and if the characteristic is $p$, we determine all possible forms of $\ell$ through an analysis of the behavior of the Frobenius endomorphism and the trace of a $C_p$-Galois extension.