Geodesic
On the Class-Number of the Maximal Real Subfield of a Cyclotomic Field
Canadian journal of mathematics, Tome 33 (1981) no. 1, pp. 55-58

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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 
Takeuchi, Hiroshi. On the Class-Number of the Maximal Real Subfield of a Cyclotomic Field. Canadian journal of mathematics, Tome 33 (1981) no. 1, pp. 55-58. doi: 10.4153/CJM-1981-006-8
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[1] 1. Ankeny, N. C., Chowla, S. and Hasse, H., On the class-number of the maximal real subfield of a cyclotomic field, J. reine angew. Math. 217 (1965), 217–220. Google Scholar

[2] 2. Lang, S. D., Note on the class-number of the maximal real subfield of a cyclotomic field, J. reine angew. Math. 290 (1977), 70–72. Google Scholar

[3] 3. Yamaguchi, I., On the class-number of the maximal real subfield of a cyclotomic field, J. reine angew. Math. 272 (1975), 217–220. Google Scholar

[4] 4. Yamaguchi, I. and Oozeki, K., On the class-number of the real quadratic field, T R U (Tokyo Rika University) Mathematics 8 (1972), 13–14. Google Scholar

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