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CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  21 May 2019

LUCA CAPUTO  and
FILIPPO A. E. NUCCIO MORTARINO MAJNO DI CAPRIGLIO Open the ORCID record for FILIPPO A. E. NUCCIO MORTARINO MAJNO DI CAPRIGLIO [Opens in a new window]
FILIPPO A. E. NUCCIO MORTARINO MAJNO DI CAPRIGLIO
Affiliation:
Univ Lyon, Université Jean Monnet Saint-Étienne, CNRS UMR 5208, Institut Camille Jordan, F-42023 Saint-Étienne, France e-mails: [email protected]; [email protected]
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Abstract

We give an algebraic proof of a class number formula for dihedral extensions of number fields of degree 2q, where q is any odd integer. Our formula expresses the ratio of class numbers as a ratio of orders of cohomology groups of units and allows one to recover similar formulas which have appeared in the literature. As a corollary of our main result, we obtain explicit bounds on the (finitely many) possible values which can occur as ratio of class numbers in dihedral extensions. Such bounds are obtained by arithmetic means, without resorting to deep integral representation theory.

MSC classification

Primary: 11R29: Class numbers, class groups, discriminants 11R20: Other abelian and metabelian extensions
Secondary: 11R34: Galois cohomology 11R37: Class field theory
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Issue 2 , May 2020 , pp. 323 - 353
Copyright
Copyright © Glasgow Mathematical Journal Trust 2019

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References

Bartel, A., Class groups in dihedral extensions – some sort of Spiegelungssatz? https://mathoverflow.net/q/73033.Google Scholar
Bartel, A., On Brauer–Kuroda type relations of S-class numbers in dihedral extensions, J. Reine Angew. Math. 668 (2012), 211–244.Google Scholar
Bartel, A. and de Smit, B., Index formulae for integral Galois modules, J. Lond. Math. Soc. (2) 88(3) (2013), 845–859. MR 3145134.Google Scholar
Boltje, R., Class group relations from Burnside ring idempotents, J. Number Theory 66(2) (1997), 291–305.Google Scholar
Brauer, R., Beziehungen zwischen Klassenzahlen von Teilkörpern eines galoisschen Körpers, Math. Nachr. 4 (1951), 158–174.CrossRefGoogle Scholar
Caputo, L., The Brauer–Kuroda formula for higher S-class numbers in dihedral extensions of number fields, Acta Arith. 151(3) (2013), 217–239.CrossRefGoogle Scholar
Cartan, H. and Eilenberg, S., Homological algebra (Princeton University Press, Princeton, NJ, 1956).Google Scholar
Dirichlet, G. L., Recherches sur les formes quadratiques à coëfficients et à indéterminées complexes. Première partie, J. Reine Angew. Math. 24 (1842), 291–371.Google Scholar
Halter-Koch, F., Einheiten und Divisorenklassen in Galois’schen algebraischen Zahlkörpern mit Diedergruppe der Ordnung 2l für eine ungerade Primzahl l, Acta Arith. 33(4) (1977), 355–364.10.4064/aa-33-4-353-364CrossRefGoogle Scholar
Heller, A. and Reiner, I., Representations of cyclic groups in ring of integers, II, Ann. Math. 77(2) (1963), 318–328.Google Scholar
Hilbert, D., Ueber den Dirichlet’schen biquadratischen Zahlkörper, Math. Ann. 45(3) (1894), 309–340.10.1007/BF01446682CrossRefGoogle Scholar
Jaulent, J.-F., Unités et classes dans les extensions métabéliennes de degré nls sur un corps de nombres algébriques, Ann. Inst. Fourier (Grenoble) 31(1) (1981), ix–x, 39–62.10.5802/aif.816CrossRefGoogle Scholar
Lang, S., Cyclotomic fields I and II, combined second edition, Graduate Texts in Mathematics, vol. 121 (Springer-Verlag, New York, 1990), with an Appendix by Karl Rubin.10.1007/978-1-4612-0987-4CrossRefGoogle Scholar
Lang, S., Algebraic number theory, Graduate Texts in Mathematics, vol. 110, 2nd edition (Springer-Verlag, New York, 1994).10.1007/978-1-4612-0853-2CrossRefGoogle Scholar
Lang, S., Algebra, Graduate Texts in Mathematics, vol. 211, 3rd edition (Springer-Verlag, New York, 2002).CrossRefGoogle Scholar
Lemmermeyer, F., Kuroda’s class number formula, Acta Arith. 66(3) (1994), 245–260.CrossRefGoogle Scholar
Lemmermeyer, F., Class groups of dihedral extensions, Math. Nachr. 278(6) (2005), 679–691.CrossRefGoogle Scholar
Moser, N., Unités et nombre de classes d’une extension galoisienne diédrale de Q, Abh. Math. Sem. Univ. Hamburg 48 (1979), 54–75.Google Scholar
Nuccio Mortarino, F. A. E. Majno di Capriglio, On Zp-extensions of real abelian number fields, PhD Thesis (University of Rome La Sapienza, 2009).Google Scholar
Serre, J.-P., Local class field theory, in Algebraic Number Theory, Proc. Instructional Conf., Brighton, 1965 (Thompson, Washington, D.C., 1967), 128–161.Google Scholar
Tate, J. T., Global class field theory, in Algebraic Number Theory, Proc. Instructional Conf., Brighton, 1965 (Thompson, Washington, D.C., 1967), 162–203.Google Scholar
Walter, C. D., A class number relation in Frobenius extensions of number fields, Mathematika 24(2) (1977), 216–225. MR 0472768.CrossRefGoogle Scholar