On the Class-number of the Maximal Real Subfield of a Cyclotomic Field

Abstract

For any square-free positive integer m, let H(m) be the class-number of the field , where ζm is a primitive m-th root of unity. We show that if m = {3(8 g + 5)}2 − 2 is a square-free integer, where g is a positive integer, then H(4 m) > 1. Similar result holds for a square-free integer m = {3(8 g +7)}2 −2, where g is a positive integer. We also show that n|H(4 m) for certain positive integers m and n.