A classification of $C_{p^n}$-Tambara fields
Noah Wisdom
TL;DR
The paper addresses the classification of field-like $C_{p^n}$-Tambara functors by introducing a separation/clarified dichotomy and showing every field-like Tambara functor is a coinduction from a clarified one. The approach reduces the problem to clarified bottom levels and analyzes characteristic not $p$ versus characteristic $p$ using Frobenius and Galois trace structures, yielding a recursive construction that builds higher-level data from lower-level pieces. The main results establish that any field-like $C_{p^n}$-Tambara functor is isomorphic to $ ext{Coind}_s^n ext{ell}$ for a clarified $C_{p^s}$-Tambara field $ ext{ell}$, with a complete classification of clarified fields in characteristic $p$ and a fixed-point behavior when the $C_p$-action is nontrivial. These findings illuminate the structure of equivariant algebra in Tambara theory and enable more tractable computations of $RO(C_{p^n})$-graded invariants in equivariant homotopy theory.
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.