Theorem 11.10 in TAN is the following simplification of Kummer’s Theorem: if is a regular odd prime, then has no solutions such that . The proof itself, however, does not explicitly appeal to the regularity of . Where is this required?
The regularity of is implicitly used when we appeal to Corollary 10.5 to show that . Essentially, we show that is principal for some ideal , but since does not divide the order of the class group, it must be that is itself principal.