Class numbers of large degree nonabelian number fields
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- by Kwang-Seob Kim and John C. Miller HTML | PDF
- Math. Comp. 88 (2019), 973-981 Request permission
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.
References
- Jacques Basmaji and Ian Kiming, A table of $A_5$-fields, On Artin’s conjecture for odd $2$-dimensional representations, Lecture Notes in Math., vol. 1585, Springer, Berlin, 1994, pp. 37–46, 122–141. MR 1322317, DOI 10.1007/BFb0074108
- Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
- B. Cais, Serre’s Conjectures (2005, revised 2009), http://math.arizona.edu/~cais/Papers/Expos/Serre05.pdf.
- Jordi Guàrdia, Jesús Montes, and Enric Nart, Higher Newton polygons in the computation of discriminants and prime ideal decomposition in number fields, J. Théor. Nombres Bordeaux 23 (2011), no. 3, 667–696 (English, with English and French summaries). MR 2861080, DOI 10.5802/jtnb.782
- Jordi Guàrdia, Jesús Montes, and Enric Nart, Newton polygons of higher order in algebraic number theory, Trans. Amer. Math. Soc. 364 (2012), no. 1, 361–416. MR 2833586, DOI 10.1090/S0002-9947-2011-05442-5
- Jordi Guàrdia, Jesús Montes, and Enric Nart, Higher Newton polygons and integral bases, J. Number Theory 147 (2015), 549–589. MR 3276340, DOI 10.1016/j.jnt.2014.07.027
- J. Guàrdia, J. Montes, and E. Nart, The Montes Package Web Page. http://www-ma4.upc.edu/~guardia/MontesAlgorithm.html
- J. Klüners and G. Malle, A Database for Number Fields. http://galoisdb.math.upb.de/home
- Kwang-Seob Kim, A construction of nonabelian simple étale fundamental groups, Ramanujan J. 35 (2014), no. 1, 111–120. MR 3258601, DOI 10.1007/s11139-014-9570-y
- A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), no. 4, 515–534. MR 682664, DOI 10.1007/BF01457454
- Jacques Martinet, Petits discriminants des corps de nombres, Number theory days, 1980 (Exeter, 1980) London Math. Soc. Lecture Note Ser., vol. 56, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 151–193 (French). MR 697261
- John C. Miller, Class numbers of totally real fields and applications to the Weber class number problem, Acta Arith. 164 (2014), no. 4, 381–398. MR 3244941, DOI 10.4064/aa164-4-4
- John C. Miller, Real cyclotomic fields of prime conductor and their class numbers, Math. Comp. 84 (2015), no. 295, 2459–2469. MR 3356035, DOI 10.1090/S0025-5718-2015-02924-X
- A. M.Odlyzko, Table 3: GRH bounds for discriminants, http://www.dtc.umn.edu/~odlyzko/unpublished/discr.bound.table3
- Georges Poitou, Sur les petits discriminants, Séminaire Delange-Pisot-Poitou, 18e année: 1976/77, Théorie des nombres, Fasc. 1, Secrétariat Math., Paris, 1977, pp. Exp. No. 6, 18 (French). MR 551335
- J.-P. Serre, Modular forms of weight one and Galois representations, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 193–268. MR 0450201
- Ken Yamamura, Maximal unramified extensions of imaginary quadratic number fields of small conductors, J. Théor. Nombres Bordeaux 9 (1997), no. 2, 405–448 (English, with English and French summaries). MR 1617407
Additional Information
- Kwang-Seob Kim
- Affiliation: Department of Mathematics, Chosun Univeristy, Gwangju 501-759, Korea
- MR Author ID: 1079146
- 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
- MR Author ID: 1074298
- Email: jmill268@jhu.edu
- 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
- © Copyright 2018 American Mathematical Society
- Journal: Math. Comp. 88 (2019), 973-981
- MSC (2010): Primary 11R29; Secondary 11Y40
- DOI: https://doi.org/10.1090/mcom/3335
- MathSciNet review: 3882291