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.

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