On the First Case of the Fermat Theorem for Cyclotomic Fields

Abstract

has no nonzero solutions in K. Let K be a finite extension of the rational number field Q (number field). One says that the first case of Fermat theorem (1CFT) holds for K and l if Eq. (1) has no integral solutions in K that are relatively prime to l. A remarkable achievement of the last few years is the proof of a problem that was standing for more than three hundred years, the FT for Q and all l > 3 (see [15]). Meanwhile, the analysis of solutions to Fermat equations in wider domains than Q, namely in its finite (or even infinite) extensions, is, apparently, still a more difficult and intriguing problem. The present paper deals with a generalization of the classical criteria of Kummer, Mirimanoff, and Vandiver for the validity of the CFT for Q and l. In brief, we prove the fulfillment of the 1CFT for L and l under the same assumptions on l that enter the corresponding classical criterion. Here L is the cyclotomic field generated over Q by the lth roots of unity. A number of authors simplified the sufficient conditions on l in the above-mentioned criteria. As a consequence, the corresponding criteria for fulfillment of the 1CFT for Q and l arose. For example, Wieferich’s criterion [14]: the 1CFT holds for Q and l if l2 does not divide 2−2. Certainly, these criteria for Q became of no importance because of the complete proof of the FT for Q by other methods. Our result implies that the 1CFT holds for L and l under the relevant conditions on l. In particular, the 1CFT holds for L and l if l2 does not divide 2 − 2. Let us state the obtained results in detail. Let ζ be a primitive lth root of unity. Let O be the ring of integers of the field L = Q(ζ), and let α = (ζ − 1)O be a unique prime ideal of O containing l. The residue field O/α can be identified with Z/lZ via the canonical isomorphism Z/lZ −→ O/α. Let (x, y, z) be a solution to Eq. (1) in O\α. Let G = G(x,y,z) be the subset of Z/lZ consisting of the classes (mod α) of the elements −x/y, −x/z, −y/x, −y/z, −z/x, −z/y. Since x+y+z ≡ x+y+z = 0 (mod α), the set G contains neither 0 nor 1. If t ∈ G, then G is exactly the set {t, 1− t, 1/t, 1/(1 − t), t/(t− 1), 1− 1/t}. Let rational numbers Bn be defined by the expansion