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Tame kernels and second regulators of number fields and their subfields

Published online by Cambridge University Press:  17 July 2013

Jerzy Browkin  and
Herbert Gangl
Jerzy Browkin
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, PL-00-956 Warsaw, [email protected]
Herbert Gangl
Affiliation:
Department of Mathematical Sciences, South Road, University of Durham, Durham, DH1 3LE, United [email protected]
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Abstract

Assuming a version of the Lichtenbaum conjecture, we apply Brauer-Kuroda relations between the Dedekind zeta function of a number field and the zeta function of some of its subfields to prove formulas relating the order of the tame kernel of a number field F with the orders of the tame kernels of some of its subfields. The details are given for fields F which are Galois over ℚ with Galois group the group ℤ/2 × ℤ/2, the dihedral group D2p; p an odd prime, or the alternating group A4. We include numerical results illustrating these formulas.

Keywords

Dedekind zeta functionsBloch-Wigner dilogarithmBloch groupsecond regulatorBrauer-Kuroda relations11R4211G55
Type
Research Article
Information
Journal of K-Theory , Volume 12 , Issue 1: Nanjing Special Issue on K-theory, number theory and geometry , August 2013 , pp. 137 - 165
Copyright
Copyright © ISOPP 2013 

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References

1.Bartel, A., de Smit, B., Index formulae for integral Galois modules, J. of London Math. Soc., to appear.Google Scholar
2.Belabas, K., Gangl, H., Generators and relations for K 2O F, K-Theory 31(3) (2004), 195–230.CrossRefGoogle Scholar
3.Bloch, S., Higher Regulators, Algebraic K-Theory, and Zeta Functions of Elliptic Curves, CRM Monograph Series 11, American Mathematical Society, Providence, Rhode Island, USA, 2000.Google Scholar
4.Bosma, W., Cannon, J., Playoust, C., The Magma algebra system. I. The user language, J. Symbolic Comp. 24 (1997), 235–265.Google Scholar
5.Brauer, R., Beziehungen zwischen Klassenzahlen von Teilkörpern eines galoisschen Körpers, Math. Nachr. 4 (1951), 158–174.Google Scholar
6.Browkin, J., Conjectures on the dilogarithm, K-Theory 3 (1989), 29–56.Google Scholar
7.Browkin, J., Brzeziński, J., Xu, Kejian, On exceptions in the Brauer-Kuroda relations, Bull. Polish Acad. Sci. Math. 59(3) (2011), 207–214.CrossRefGoogle Scholar
8.Browkin, J., Schinzel, A., On Sylow 2-subgroups of K 2O F for quadratic number fields F, Journ. Reine Angew. Math. 331 (1982), 104–113.Google Scholar
9.Burns, D., de Jeu, R., Gangl, H., On special values in higher algebraic K-theory and the Lichtenbaum-Gross conjecture, Advances in Math. 230(3) (2012), 1502–1529.CrossRefGoogle Scholar
10.Cathelineau, J.-L., Remarques sur les différentielles des polylogarithmes uniformes, Ann. Inst. Fourier, Grenoble 46(5) (1996), 1327–1347.Google Scholar
11.Cohen, H., A Course in Computational Algebraic Number Theory (3rd corrected printing), Graduate Texts in Mathematics 138, Springer Verlag, Berlin, 1996.Google Scholar
12.Fröhlich, A., Artin root numbers and normal integral bases for quaternion fields, Inventiones math. 17 (1972), 143–166.Google Scholar
13.The GAP group, GAP - Groups, Algorithms, and Programming, Version 4.4.12; 2008 (http://www.gap-system.org).Google Scholar
14.Hasse, H., Über die Klassenzahl Abelscher Zahlkörper, Akademie-Verlag, Berlin, 1952.CrossRefGoogle Scholar
15.Kuroda, S., Über die Klassenzahlen algebraischer Zahlkörper, Nagoya Math. J. 1 (1950), 1–10.Google Scholar
16.Milnor, J., Introduction to algebraic K-theory, Princeton University Press, Princeton, N.J., 1971.Google Scholar
17.Narkiewicz, W., Elementary and Analytic Theory of Algebraic Numbers, 3rd ed.Springer, Berlin, 2004.CrossRefGoogle Scholar
18.The PARI-group, PARI/GP, versions 2.1-2.4, Bordeaux, url http://pari.math.ubordeaux.fr.Google Scholar
19.Ribenboim, P., Classical Theory of Algebraic Numbers, Springer, New York, 2001.Google Scholar
20.Skałba, M., Generalization of Thue's theorem and computation of the group K 2O F, J. Number Theory 46 (1994), 303–322.Google Scholar
21.Tate, J., Appendix to The Milnor ring of a global field, in: Algebraic K-theory, II: ‘Classical’ algebraic K-theory and connections with arithmetic (Proc. Conf. Seattle, Wash., Battelle Memorial Inst., 1972), Lecture Notes in Math. 342 (1973), 349–446.Google Scholar
22.Wiles, A., The Iwasawa conjecture for totally real fields, Ann. of Math. 131 (1990), 493–540.CrossRefGoogle Scholar
23.Haiyan, Zhou, The tame kernel of multiquadratic number fields, Comm. Algebra 37(2) (2009), 630–638.Google Scholar
24.Haiyan, Zhou, Higher class numbers in extensions of number fields, (submitted) (2012).Google Scholar