A note on the quaternion group as Galois group

Ware, Roger

doi:10.1090/S0002-9939-1990-0998741-4

A note on the quaternion group as Galois group
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by Roger Ware

Proc. Amer. Math. Soc. 108 (1990), 621-625

DOI: https://doi.org/10.1090/S0002-9939-1990-0998741-4

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Abstract:

The occurrence of the quaternion group as a Galois group over certain fields is investigated. A theorem of Witt on quaternionic Galois extensions plays a key role.

References

Jón Kr. Arason, Richard Elman, and Bill Jacob, Rigid elements, valuations, and realization of Witt rings, J. Algebra 110 (1987), no. 2, 449–467. MR 910395, DOI 10.1016/0021-8693(87)90057-3
A. Fröhlich, Orthogonal representations of Galois groups, Stiefel-Whitney classes and Hasse-Witt invariants, J. Reine Angew. Math. 360 (1985), 84–123. MR 799658, DOI 10.1515/crll.1985.360.84
Bill Jacob and Roger Ware, A recursive description of the maximal pro-$2$ Galois group via Witt rings, Math. Z. 200 (1989), no. 3, 379–396. MR 978598, DOI 10.1007/BF01215654
T. Y. Lam, The algebraic theory of quadratic forms, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Reading, Mass., 1973. MR 0396410
Ján Mináč, Quaternion fields inside Pythagorean closure, J. Pure Appl. Algebra 57 (1989), no. 1, 79–82. MR 984048, DOI 10.1016/0022-4049(89)90029-7
Jean-Pierre Serre, L’invariant de Witt de la forme $\textrm {Tr}(x^2)$, Comment. Math. Helv. 59 (1984), no. 4, 651–676 (French). MR 780081, DOI 10.1007/BF02566371
Roger Ware, When are Witt rings group rings? II, Pacific J. Math. 76 (1978), no. 2, 541–564. MR 568320
Roger Ware, 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), no. 1, 95–107. MR 716238, DOI 10.1016/0022-4049(83)90042-7

Konstruktion von galoisschen Körpen der Charakteristik

zu vorgegebener Gruppe der Ordnung

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