Algebra
Extensions with Galois group 2+S4D8 in characteristic 3
Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 611-614.

For K a field of characteristic 3 we give explicitly the whole family of Galois extensions of K with Galois group 2+S4D8 and determine the discriminant of such an extension. In the case when K is the field of fractions of a formal power series ring in 3 variables, this result is interesting in the context of Abhyankar's Normal Crossings Local Conjecture.

Pour K un corps de caractéristique 3 nous donnons explicitement la famille complète d'extensions de K à groupe de Galois 2+S4D8 et déterminons le discriminant d'une telle extension. Dans le cas où K est le corps de fractions d'un anneau de séries de puissances formelles en 3 variables, ce résultat est intéressant dans le contexte de la Conjecture Locale de Croisements Normaux d'Abhyankar.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2008.04.010
Crespo, Teresa 1; Hajto, Zbigniew 2

1 Departament d'Àlgebra i Geometria, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain
2 Theoretical Computer Science Department, Jagiellonian University, ul. Gronostajowa 3, 30387 Kraków, Poland
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Crespo, Teresa; Hajto, Zbigniew. Extensions with Galois group $ {2}^{+}{S}_{4}\ast {D}_{8}$ in characteristic 3. Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 611-614. doi : 10.1016/j.crma.2008.04.010. http://www.numdam.org/articles/10.1016/j.crma.2008.04.010/

[1] Abhyankar, S.S. Galois embeddings for linear groups, Trans. Amer. Math. Soc., Volume 352 (2000), pp. 3881-3912

[2] Abhyankar, S.S. Resolution of singularities and modular Galois theory, Bull. Amer. Math. Soc., Volume 38 (2001), pp. 131-169

[3] Crespo, T.; Hajto, Z. On vectorial polynomials and coverings in characteristic 3, Proc. Amer. Math. Soc., Volume 134 (2006), pp. 23-29

[4] Harbater, D.; van der Put, M.; Guralnick, R. Valued fields and covers in characteristic p, Fields Inst. Commun., Volume 32 (2002), pp. 175-204

[5] Lam, T.Y. The Algebraic Theory of Quadratic Forms, Benjamin-Cummings Publ. Co., Reading, MA, 1973

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