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