Imaginary multiquadratic number fields with class group of exponent 3 and 5

Klüners, Jürgen; Komatsu, Toru

doi:10.1090/mcom/3609

Imaginary multiquadratic number fields with class group of exponent $3$ and $5$
HTML articles powered by AMS MathViewer

by Jürgen Klüners and Toru Komatsu HTML | PDF

Math. Comp. 90 (2021), 1483-1497 Request permission

Abstract:

In this paper we obtain a complete list of imaginary $n$-quadratic fields with class groups of exponent $3$ and $5$ under extended Riemann hypothesis for every positive integer $n$ where an $n$-quadratic field is a number field of degree $2^n$ represented as the composite of $n$-quadratic fields.

References

Eric Bach and Jonathan Sorenson, Explicit bounds for primes in residue classes, Math. Comp. 65 (1996), no. 216, 1717–1735. MR 1355006, DOI 10.1090/S0025-5718-96-00763-6
A. Baker, Linear forms in the logarithms of algebraic numbers. I, II, III, Mathematika 13 (1966), 204–216; ibid. 14 (1967), 102–107; ibid. 14 (1967), 220–228. MR 220680, DOI 10.1112/s0025579300003843
Jeff Bezanson, Alan Edelman, Stefan Karpinski, and Viral B. Shah, Julia: a fresh approach to numerical computing, SIAM Rev. 59 (2017), no. 1, 65–98. MR 3605826, DOI 10.1137/141000671
David W. Boyd and H. Kisilevsky, On the exponent of the ideal class groups of complex quadratic fields, Proc. Amer. Math. Soc. 31 (1972), 433–436. MR 289454, DOI 10.1090/S0002-9939-1972-0289454-4
Ezra Brown and Charles J. Parry, The imaginary bicyclic biquadratic fields with class-number $1$, J. Reine Angew. Math. 266 (1974), 118–120. MR 340219, DOI 10.1515/crll.1974.266.118
Andreas-Stephan Elsenhans, Jürgen Klüners, and Florin Nicolae, Imaginary quadratic number fields with class groups of small exponent, Acta Arith. 193 (2020), no. 3, 217–233. MR 4071803, DOI 10.4064/aa180220-20-3
Claus Fieker, William Hart, Tommy Hofmann, and Fredrik Johansson, Nemo/Hecke: computer algebra and number theory packages for the Julia programming language, ISSAC’17—Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2017, pp. 157–164. MR 3703682, DOI 10.1145/3087604.3087611
Albrecht Fröhlich, Central extensions, Galois groups, and ideal class groups of number fields, Contemporary Mathematics, vol. 24, American Mathematical Society, Providence, RI, 1983. MR 720859, DOI 10.1090/conm/024
D. R. Heath-Brown, Imaginary quadratic fields with class group exponent 5, Forum Math. 20 (2008), no. 2, 275–283. MR 2394923, DOI 10.1515/FORUM.2008.014
Seok-Won Jung and Soun-Hi Kwon, Determination of all imaginary bicyclic biquadratic number fields of class number $3$, Bull. Korean Math. Soc. 35 (1998), no. 1, 83–89. MR 1608982
Franz Lemmermeyer, Kuroda’s class number formula, Acta Arith. 66 (1994), no. 3, 245–260. MR 1276992, DOI 10.4064/aa-66-3-245-260
The LMFDB Collaboration, The L-functions and Modular Forms Database, http://www.lmfdb.org (2019)
G. Malle, A Database for Class Groups of Number Fields, https://service.mathematik.uni-kl.de/\~numberfieldtables/ (2011).
L. Rédei, Arithmetischer Beweis des Satzes über die Anzahl der durch vier teilbaren Invarianten der absoluten Klassengruppe im quadratischen Zahlkörper, J. Reine Angew. Math. 171 (1934), 55–60 (German). MR 1581419, DOI 10.1515/crll.1934.171.55
H. M. Stark, A complete determination of the complex quadratic fields of class-number one, Michigan Math. J. 14 (1967), 1–27. MR 222050
Kôji Uchida, Imaginary abelian number fields with class number one, Tohoku Math. J. (2) 24 (1972), 487–499. MR 321904, DOI 10.2748/tmj/1178241490
Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575, DOI 10.1007/978-1-4612-1934-7
Mark Watkins, Class numbers of imaginary quadratic fields, Math. Comp. 73 (2004), no. 246, 907–938. MR 2031415, DOI 10.1090/S0025-5718-03-01517-5
Ken Yamamura, The determination of the imaginary abelian number fields with class number one, Math. Comp. 62 (1994), no. 206, 899–921. MR 1218347, DOI 10.1090/S0025-5718-1994-1218347-3