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On generalized Mersenne Primes and class-numbers of equivalent quadratic fields and cyclotomic fields

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Abstract

In this paper we define equivalent quadratic fields and prove that generalized Mersenne primes generate a family of infinitely many equivalent quadratic fields with equivalent index \(2\) and whose class numbers are divisible by 3. We also prove that the class-number of the cyclotomic field \(\mathbb {Q}\big ( \zeta _m \big )\), where \(m\in \mathbb {N}\) and \(\zeta _m\) is a primitive \(m\)-th root of unity, is divisible by a certain integer \(g\).

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Acknowledgments

The first author acknowledges UGC for JRF Fellowship (No. GU/UGC/VI(3)/JRF/2012/2985).

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  1. Department of Mathematics, Gauhati University, Guwahati, 781014, India

    Azizul Hoque & Helen K. Saikia

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  1. Azizul Hoque
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  2. Helen K. Saikia
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Correspondence to Azizul Hoque.

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Hoque, A., Saikia, H.K. On generalized Mersenne Primes and class-numbers of equivalent quadratic fields and cyclotomic fields. SeMA 67, 71–75 (2015). https://doi.org/10.1007/s40324-014-0027-4

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  • DOI: https://doi.org/10.1007/s40324-014-0027-4

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