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Real cyclotomic fields of prime conductor and their class numbers


Author: John C. Miller
Journal: Math. Comp. 84 (2015), 2459-2469
MSC (2010): Primary 11R29, 11R18; Secondary 11R80, 11Y40
DOI: https://doi.org/10.1090/S0025-5718-2015-02924-X
Published electronically: February 5, 2015
MathSciNet review: 3356035
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Abstract: Surprisingly, the class numbers of cyclotomic fields have only been determined for fields of small conductor, e.g., for prime conductors up to 67, due to the problem of finding the ``plus part,'' i.e., the class number of the maximal real subfield. Our results have improved the situation. We prove that the plus part of the class number is 1 for prime conductors between 71 and 151. Also, under the assumption of the generalized Riemann hypothesis, we determine the class number for prime conductors between 167 and 241. This technique generalizes to any totally real field of moderately large discriminant, allowing us to confront a large class of number fields not previously treatable.


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Additional Information

John C. Miller
Affiliation: Department of Mathematics, Rutgers University, Hill Center for the Mathematical Sciences, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019
Email: jcmiller@math.rutgers.edu

DOI: https://doi.org/10.1090/S0025-5718-2015-02924-X
Received by editor(s): October 28, 2013
Received by editor(s) in revised form: December 13, 2013
Published electronically: February 5, 2015
Article copyright: © Copyright 2015 American Mathematical Society