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Mathematics of Computation

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Class numbers of large degree nonabelian number fields


Authors: Kwang-Seob Kim and John C. Miller
Journal: Math. Comp. 88 (2019), 973-981
MSC (2010): Primary 11R29; Secondary 11Y40
DOI: https://doi.org/10.1090/mcom/3335
Published electronically: April 27, 2018
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Abstract: If a number field has a large degree and discriminant, the computation of the class number becomes quite difficult, especially without the assumption of GRH. In this article, we will unconditionally show that a certain nonabelian number field of degree 120 has class number one. This field is the unique $ A_5 \times C_2$ extension of the rationals that is ramified only at 653 with ramification index 2. It is the largest degree number field unconditionally proven to have class number 1.

The proof uses the algorithm of Guàrdia, Montes, and Nart to calculate an integral basis and then finds integral elements of small prime power norm to establish an upper bound for the class number; further algebraic arguments prove the class number is 1. It is possible to apply these techniques to other nonabelian number fields as well.


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Additional Information

Kwang-Seob Kim
Affiliation: Department of Mathematics, Chosun Univeristy, Gwangju 501-759, Korea
Email: kwang12@chosun.ac.kr

John C. Miller
Affiliation: Department of Applied Mathematics & Statistics, Johns Hopkins University, 100 Whitehead Hall, 3400 North Charles Street, Baltimore, Maryland 21218
Email: jmill268@jhu.edu

DOI: https://doi.org/10.1090/mcom/3335
Received by editor(s): August 1, 2016
Received by editor(s) in revised form: April 30, 2017, October 25, 2017, and October 26, 2017
Published electronically: April 27, 2018
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society