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Decomposition of Witt Rings and Galois Groups

Published online by Cambridge University Press:  20 November 2018

Ján Mináč  and
Tara L. Smith
Ján Mináč
Affiliation:
Department of Mathematics University of Western Ontario London, Ontario N6A 5B7
Tara L. Smith
Affiliation:
Department of Mathematical Sciences University of Cincinnati Cincinnati, Ohio 45221-0025 U.S.A.
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Abstract

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To each field F of characteristic not 2, one can associate a certain Galois group 𝒢F, the so-called W-group of F, which carries essentially the same information as the Witt ring W(F) of F. In this paper we show that direct products of Witt rings correspond to free products of these Galois groups (in the appropriate category), group ring construction of Witt rings corresponds to semidirect products of W-groups, and the basic part of W(F) is related to the center of 𝒢F. In an appendix we provide a complete list of Witt rings and corresponding w-groups for fields F with |Ḟ/Ḟ2| ≤ 16.

Keywords

11E81
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 6 , 01 December 1995 , pp. 1274 - 1289
Copyright
Copyright © Canadian Mathematical Society 1995

References

[AEJ:1984] Arason, J., Elman, R. and Jacob, B., The graded Witt ring and Galois cohomology, CMS Conf. Proc. 4(1984), 17–50.Google Scholar
[AEJ:1987] Arason, J., Rigid elements, valuations, and realization of Witt rings,J. Algebra 110(1987), 449–467.Google Scholar
[Be: 1978] Berman, L., The Kaplansky Radical and Values of Binary Quadratic Forms over Fields, Ph.D. Thesis, University of California, Berkeley, California, 1978.Google Scholar
[CMa:1982] Carson, A. and Marshall, M., Decomposition of Witt rings, Canad. J. Math 34(1982), 1276–1302.Google Scholar
[J:1981] Jacob, B., On the structure of Pythagorean fields, J. Algebra 68(1981), 247–267.Google Scholar
[JWr:1989] Jacob, B. and Ware, R., A recursive description of the maximal pro-2-Galois group via Witt rings, Math. Z. 200(1989), 379–396.Google Scholar
[La: 1973] Lam, T.Y., The Algebraic Theory of Quadratic Forms, W. A. Benjamin, Reading, Massachusetts, 1973, Second printing with revisions, 1980.Google Scholar
[Ma: 1980] Marshall, M., Abstract Witt Rings, Queen's Papers in Pure and Appl. Math. 57, Queen's University, Kingston, Ontario, Canada, 1980.Google Scholar
[MiSm:1993] Minâc, J., Smith, L., W-groups and values of binary forms, J. Pure Appl. Algebra 87(1993), 61–78.Google Scholar
[MiSm:pre] Smith, L., Quotients and subgroups of W-groups, preprint.Google Scholar
[MiSp:1990] Minâc, J. and Spira, M., Formally real fields, pythagorean fields, C-fields, and W-groups, Math. Z. 205(1990), 519–530.Google Scholar
[MiSp:1995] , Witt rings and Galois groups, Annals of Math., to appear.Google Scholar
[N:1971] Neukirch, J., Freie Produkte proendlicher Gruppen und ihre Kohomologie, Arch. Math 22(1971), 337–357.Google Scholar
[Pf:1966] Pfister, A., Quadratische Formen in beliebigen Korpern, Invent. Math. 1(1966), 116–132.Google Scholar
[Pn:1966] Pontrjagin, L., Topological Groups, Gordon and Breach, 1966.Google Scholar
[Sc:1985] Scharlau, W., Quadratic and Hermitian Forms, Grundlehren Math. Wiss. 270, Springer-Verlag, Berlin, 1985.Google Scholar
[Se: 1965] Serre, J.-P., Cohomologie Galoisienne, Lecture Notes in Math. 5, Springer-Verlag, New York, Berlin, 1965. Cinquième édition, révisée et complétée, 1994.Google Scholar
[Sm:1988] Smith, T.L., Some 2-Groups Arising in Quadratic Form Theory and Their Generalizations, Ph.D. Thesis, University of California, Berkeley, California, 1988.Google Scholar
[Sp:1987] Spira, M., Witt Rings and Galois Groups, Ph.D. Thesis, University of California, Berkeley, California, 1987.Google Scholar
[Wr:1979] Ware, R., Quadratic forms and pro-finite 2-groups, J. Algebra 58(1979), 227–237.Google Scholar
[Wr:1981] , Valuation rings and rigid elements infields, Canad. J. Math 33(1981), 1334.1355.Google Scholar
[Wr:1983] , Quadratic forms and pro-2-groups II: The galois group of the pythagorean closure of a formally real field, J. Pure Appl. Algebra 30(1983), 95–107. [Wr:1985] , Quadratic forms and pro-2-groups III, Comm. Algebra 13(1985), 1713.1736.Google Scholar
[Wi:1936] Witt, E., Konstruktion von galoisschen Körpern der Charakteristik p zu vorgegebener Gruppe der Ordnung pf, J. Reine Angew. Math. 174(1936), 237–245.Google Scholar