NOTE ON THE RELATIVE CLASS NUMBER OF THE 7ⁿ th CYCLOTOMIC FIELD

Abstract

Let p be an odd prime number, and a prime number with ≠ p. Let h_n⁻ be the relative class number of the pⁿ⁺¹th cyclotomic field Q(ζ_pⁿ⁺¹). When p = 3 or 5, it is known that h_n⁻ for any n ≥ 0 if ² 1 mod 9 (respectively ⁸ 1 mod 100). We show some corresponding results for the case p = 7.