A classification of Cpn-Tambara fields

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 ℓ such that $\ell (C_{p^s}/e)$ is a field. If this field has characteristic other than p, we observe that ℓ must be a fixed-point Tambara functor, and if the characteristic is p, we determine all possible forms of ℓ through an analysis of the behavior of the Frobenius endomorphism and the trace of a $C_p$-Galois extension.