## Analyze a proof of a simplification of Kummer’s Theorem

Theorem 11.10 in TAN is the following simplification of Kummer’s Theorem: if $p$ is a regular odd prime, then $x^p + y^p = z^p$ has no solutions such that $p|xyz$. The proof itself, however, does not explicitly appeal to the regularity of $p$. Where is this required?

The regularity of $p$ is implicitly used when we appeal to Corollary 10.5 to show that $[x+\zeta y] = [\delta^p]$. Essentially, we show that $A^p$ is principal for some ideal $A$, but since $p$ does not divide the order of the class group, it must be that $A$ is itself principal.