Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T16:56:26.014Z Has data issue: false hasContentIssue false
  • English
  • Français

Brauer Group Analogues of Results Relating the Witt Ring to Valuations and Galois Theory

Published online by Cambridge University Press:  20 November 2018

Yoon Sung Hwangk  and
Bill Jacob
Yoon Sung Hwangk
Affiliation:
Korea University, Seoul136-701, Korea
Bill Jacob
Affiliation:
University of California-Santa Barbara, Santa Barbara, California 93106, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let F be a field of characteristic different from p containing a primitive p-th root of unity. This paper studies the cup product pairing Hl(F, p) x Hl(F, p) → H2(F, p) and its relationship to valuation theory and Galois theory. Sufficient conditions on the pairing which guarantee the existence of a valuation on the field are described. In the non p-adic case these results provide a converse to the well-known structure theory in this situation. In the p-adic case, the pairing is described using the notion of "relative rigidity". These results are analogues of results in quadratic form theory developed in the past decade, which cover the special case p = 2. Applications to the maximal pro-p Galois group of F are also described.

Keywords

1216
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 3 , 01 June 1995 , pp. 527 - 543
Copyright
Copyright © Canadian Mathematical Society 1995

References

[AEJ] Arason, J., Elman, R. and Jacob, B., Rigid Elements, valuations, and realization of Witt rings, J. Algebra 110(1987), 449–467.Google Scholar
[B] Becker, E., Formal-reelle Kôrper mit streng-auflösbarer absoluter Galoisgruppe, Math. Ann. 238(1978), 203–206.Google Scholar
[J] Jacob, B., Quadratic forms over Dyadic Valued Fields II, Relative Rigidity and Galois Cohomology, J. Algebra 148(1992), 162–202.Google Scholar
[JWd] Jacob, B. and Wadsworth, A., A New Construction of Nuncrossed Product Algebras, Trans. Amer. Math. Soc. 293(1986), 693–721.Google Scholar
[JWr 1] 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
[JWr 2] Jacob, B., Realizing dyadic factors of elementary type Witt rings and pro-2 Galois groups, Math. Z. 208(1991), 193–208.Google Scholar
[L] Lam, T.-Y., Algebraic Theory of Quadratic Forms, W. A. Benjamin Inc., 1973.Google Scholar
[K] Kato, K., A generalization of local class field theory by using K-groups, I, II, J. Fac. Sci. Univ. Tokyo Sect. IA Math. (2) 26(1979), 303–376. (3) 27(1980), 603–683.Google Scholar
[M 1] Marshall, M., Abstract Witt Rings, Queen's Papers in Pure and Appl. Math. 57, 1980.Google Scholar
[M 2] Marshall, M., Classification of finitely generated Witt rings which are strongly representational, preprint.Google Scholar
[MY] Marshall, M. and Yucas, J., Linked quaternionic mappings and their associated Witt rings, Pacific J. Math. 95(1981), 411–426.Google Scholar
[Me] Merkmjev, A.S., K2 and the Brauer Group, Contemp. Math. 55(1986), 529–547.Google Scholar
[W 1] Ware, R., Quadratic Forms and Profinite 2-groups, J. Algebra 58(1979), 227–237.Google Scholar
[W 2] Ware, R., Valuation Rings and rigid elements infields, Canad. J. Math. 33(1981), 1338–1355.Google Scholar