Quartic fields with large class numbers. (English) Zbl 1448.11201

In this paper, the authors show that there exist infinitely many quartic number fields \(K\) with large class numbers such that \(K/\mathbb Q\) is a Galois extension whose Galois group is isomorphic to a given finite group. P. J. Cho and H. H. Kim [J. Théor. Nombres Bordx. 24, No. 3, 583–603 (2012; Zbl 1275.11145)] proved that there are infinitely many totally real cyclic extensions over \(\mathbb Q\) of degree 4 with large class numbers. In this paper, the authors consider all the other cases of quartic Galois extensions (totally real biquadratic extensions, biquadratic CM-fields and quartic cyclic CM-fields).


11R29 Class numbers, class groups, discriminants
11R16 Cubic and quartic extensions


Zbl 1275.11145
Full Text: DOI Euclid


[1] P. J. Cho, “The strong Artin conjecture and large class numbers”, Q. J. Math. 65:1 (2014), 101-111. \xoxMR3179652 · Zbl 1303.11124
[2] P. J. Cho \andword H. H. Kim, “Dihedral and cyclic extensions with large class numbers”, J. Théor. Nombres Bordeaux 24:3 (2012), 583-603. \xoxMR3010630 · Zbl 1275.11145
[3] P. J. Cho \andword H. H. Kim, “Application of the strong Artin conjecture to the class number problem”, Canad. J. Math. 65:6 (2013), 1201-1216. \xoxMR3121669 · Zbl 1283.11155
[4] S. Chowla, “On the \(k\)-analogue of a result in the theory of the Riemann zeta function”, Math. Z. 38:1 (1934), 483-487. \xoxMR1545462 · JFM 60.0153.02
[5] S. Chowla, “On the class-number of the corpus \(P(\sqrt{-k})\)”, Proc. Nat. Inst. Sci. India 13 (1947), 197-200. \xoxMR27303
[6] T. W. Cusick, “Lower bounds for regulators”, \ppword 63-73 \inword Number theory (Noordwijkerhout, 1983), Lecture Notes in Math. 1068, Springer, Berlin, 1984. \xoxMR756083 · Zbl 0549.12003
[7] R. C. Daileda, “Non-abelian number fields with very large class numbers”, Acta Arith. 125:3 (2006), 215-255. \xoxMR2276192 · Zbl 1158.11044
[8] W. Duke, “Extreme values of Artin \(L\)-functions and class numbers”, Compositio Math. 136:1 (2003), 103-115. \xoxMR1966783 · Zbl 1013.11072
[9] W. Duke, “Number fields with large class group”, \ppword 117-126 \inword Number theory, CRM Proc. Lecture Notes 36, Amer. Math. Soc., Providence, RI, 2004. \xoxMR2076589
[10] K. Hardy, R. H. Hudson, D. Richman, K. S. Williams\serialcomma \andword N. M. Holtz, “Calculation of the class numbers of imaginary cyclic quartic fields”, Math. Comp. 49:180 (1987), 615-620. \xoxMR906194 · Zbl 0624.12004
[11] R. H. Hudson \andword K. S. Williams, “The integers of a cyclic quartic field”, Rocky Mountain J. Math. 20:1 (1990), 145-150. \xoxMR1057983 · Zbl 0707.11078
[12] T. Kubota, “Über die Beziehung der Klassenzahlen der Unterkörper des bizyklischen biquadratischen Zahlkörpers”, Nagoya Math. J. 6 (1953), 119-127. \xoxMR59960 · Zbl 0053.21902
[13] T. Kubota, “Über den bizyklischen biquadratischen Zahlkörper”, Nagoya Math. J. 10 (1956), 65-85. \xoxMR83009 · Zbl 0074.03001
[14] H. L. Montgomery \andword P. J. Weinberger, “Real quadratic fields with large class number”, Math. Ann. 225:2 (1977), 173-176. \xoxMR427271 · Zbl 0325.12001
[15] Y. Morita, A. Umegaki\serialcomma \andword Y. Umegaki, “Bicubic number fields with large class numbers”, \inword Various aspects of multiple zeta functions – in honor of Professor Kohji Matsumoto’s 60th birthday, Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo, 2020.
[16] M. Nair, “Power free values of polynomials”, Mathematika 23:2 (1976), 159-183. \xoxMR429801 · Zbl 0349.10039
[17] Y. Rikuna, “Explicit constructions of generic polynomials for some elementary groups”, \ppword 173-194 \inword Galois theory and modular forms, Dev. Math. 11, Kluwer Acad. Publ., Boston, MA, 2004. \xoxMR2059763
[18] J. H. Silverman, “An inequality relating the regulator and the discriminant of a number field”, J. Number Theory 19:3 (1984), 437-442. \xoxMR769793 · Zbl 0552.12003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.