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

Let p be an integer and let H(p) be the class-number of the field where ζ p is a primitive p-th root of unity and Q is the field of rational numbers. It has been proved in [1] that if p = (2qn)2 + 1 is a prime, where q is a prime and n > 1 an integer, then H(p) > 1. Later, S. D. Lang [2] proved the same result for the prime number p = ((2n + 1)q)2 + 4, where q is an odd prime and n ≧ 1 an integer. Both results have been obtained in the case p ≡ 1 (mod 4). In this paper we shall prove the similar results for a certain prime number p ≡ 3 (mod 4). We designate by h(p) the class-number of the real quadratic field