Class Numbers and a Generalized Fermat Theorem

The purpose of this paper is twofold. In the first section we provide sufficient conditions for the divisibility of the class number of certain algebraic number fields by prime powers and applications thereof. We were initially inspired in this regard by Watabe [ 111, wherein some of the results are false thereby leading to the above. In the second section we generalize Fermat’s well-known “two-square theorem.” We then apply the results of the first section to the latter result to obtain a connection between the existence of an integer solution to the 2 pth cyclotomic polynomial modulo n and the divisibility of the class number of Q(E& (for all primes q dividing n) by p-powers and the divisibility of the class number of Q(E,, fl) by 2, where q* = (-l)‘“-“‘z q.