Extensions with Galois group $ {2}^{+}{S}_{4}\ast {D}_{8}$ in characteristic 3

Teresa Crespo; Zbigniew Hajto

doi:10.1016/j.crma.2008.04.010

Algebra

Extensions with Galois group

2^{+} S_{4} * D_{8}

in characteristic 3

Presented by: Jean-Pierre Serre

Teresa Crespo ¹; Zbigniew Hajto 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

Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 611-614.

Abstract
Résumé

For K a field of characteristic 3 we give explicitly the whole family of Galois extensions of K with Galois group $2^{+} S_{4} * D_{8}$ 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 $2^{+} S_{4} * D_{8}$ 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: 2007-10-05
Accepted: 2008-04-14
Published online: 2008-05-20

DOI: 10.1016/j.crma.2008.04.010

Author's affiliations:

Teresa Crespo ¹; Zbigniew Hajto 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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     author = {Teresa Crespo and Zbigniew Hajto},
     title = {Extensions with {Galois} group $ {2}^{+}{S}_{4}\ast {D}_{8}$ in characteristic 3},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {611--614},
     publisher = {Elsevier},
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     number = {11-12},
     year = {2008},
     doi = {10.1016/j.crma.2008.04.010},
     language = {en},
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Teresa Crespo; Zbigniew Hajto. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.04.010/

References
Cited by

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

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

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

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

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