×

Twisted multiplicative field invariants, Noether’s problem, and Galois extensions. (English) Zbl 0702.12002

The author considers multiplicative invariant fields \(F(Q)^ G\) (G finite group, Q a G-lattice; F(Q) the fraction field of the group algebra F[G], where F(Q) is provided with the natural G-action). For certain lattices Q, \(F(Q)^ G\) is stably isomorphic to \(F(G')\) with \(G'\) a split extension of G with an abelian kernel. The main part of this paper is devoted to showing that \(F(G')\) can be considered as an \(\alpha\)-twisted multiplicative invariant field of G.
This work is a natural continuation of the author’s previous work. E.g. \(\alpha\)-twisted groups can be considered as a way of describing the invariant fields of reductive algebraic groups. The embedding problem is related to \(\alpha\)-twisted multiplicative invariant fields. Unirationality of certain \(\alpha\)-twisted multiplicative invariant fields is shown to be equivalent to the existence of solutions for the embedding problem, thus extending the existing number-theoretic results.
Reviewer: G.Molenberghs

MSC:

12F10 Separable extensions, Galois theory
20C10 Integral representations of finite groups
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Albert, A. A., Modern Higher Algebra (1937), Univ. of Chicago Press: Univ. of Chicago Press Chicago · Zbl 0017.29201
[3] Brown, K., Cohomology of Groups (1982), Springer-Verlag: Springer-Verlag New York/Berlin
[4] Borel, A., Linear Algebraic Groups (1969), Benjamin: Benjamin New York · Zbl 0186.33201
[5] (Cassels, J. W.S; Frohlich, A., Algebraic Number Theory (1967), Thompson: Thompson Washington, DC) · Zbl 0153.07403
[6] Colliot-Thélène, J.-L; Sansuc, J.-J, \(La R\) èquivalence sur les tores, Ann. Sci. École Norm. Sup (4), 10, 177-230 (1977)
[7] DeMeyer, F.; Ingraham, E., Separable algebras over commutative rings, (Lecture Notes in Mathematics, Vol. 181 (1971), Springer-Verlag: Springer-Verlag Berlin/New York) · Zbl 0215.36602
[9] Iwasawa, K., On solvable extensions of algebraic number fields, Ann. of Math., 58, 548-572 (1953) · Zbl 0051.26602
[10] Jansen, U., Über Galoisgruppen lokaler Korper, Invent. Math., 70, 53-69 (1982) · Zbl 0534.12009
[11] Jacobson, N., (Basic Algebra I (1974), Freeman: Freeman San Francisco) · Zbl 0284.16001
[12] Kuyk, W., Generic approach to the Galois embedding and extension problem, J. Algebra, 9, 393-407 (1968) · Zbl 0183.04005
[13] Knus, M. A.; Ojanguren, M., Théorie de la Descente et Algèbres d’Azumaya, (Lecture Notes in Mathematics, Vol. 389 (1974), Springer-Verlag: Springer-Verlag Berlin/New York) · Zbl 0284.13002
[14] Lang, S., Diophantine Geometry (1962), Wiley: Wiley New York · Zbl 0115.38701
[15] Lenstra, H. W., Rational functions invariant under a finite abelian group, Invent. Math., 25, 299-325 (1974) · Zbl 0292.20010
[17] Neukirch, J., On solvable number fields, Invent. Math., 53, 135-164 (1979) · Zbl 0447.12008
[18] Neukirch, J., Extensions of number fields with solvable Galois group, (Seminar on Number Theory. Seminar on Number Theory, Paris 1979-1980. Seminar on Number Theory. Seminar on Number Theory, Paris 1979-1980, Progress in Mathematics, 12 (1981), Birkhäuser: Birkhäuser Boston), 223-236
[19] Saltman, D. J., Generic Galois extensions and problems in field theory, Adv. in Math., 43, 250-283 (1982) · Zbl 0484.12004
[20] Saltman, D. J., Retract rational fields and cyclic Galois extensions, Israel J. Math., 47, 165-215 (1984) · Zbl 0546.14013
[21] Saltman, D. J., Groups acting on fields: Noether’s problem, (Group Actions on Rings. Group Actions on Rings, Contemporary Mathematics, No. 43 (1985), Amer. Math. Soc: Amer. Math. Soc Providence, RI) · Zbl 0568.12013
[22] Saltman, D. J., Noether’s problem over algebraically closed fields, Invent. Math., 77, 71-84 (1984) · Zbl 0546.14014
[23] Saltman, D. J., Multiplicative field invariants, J. Algebra, 106, 221-238 (1987) · Zbl 0622.13002
[26] Saltman, D. J., Azumaya algebras with involution, J. Algebra, 52, 526-539 (1978) · Zbl 0382.16003
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.