Polynomials over $Q$ solving an embedding problem

Vila, Nuria

doi:10.5802/aif.1010

Polynomials over

Q

solving an embedding problem

Vila, Nuria

Annales de l'Institut Fourier, Tome 35 (1985) no. 2, pp. 79-82.

Résumé
Abstract

Les corps de décomposition des polynômes construits dans E. Nart and the author in J. Number Theory 16, (1983), 6–13, Th. 2.1, dont le groupe de Galois est isomorphe au groupe alterné $A_{n}$ , peuvent être plongés dans toute extension centrale de $A_{n}$ si et seulement si $n \equiv 0$ mod. 8, ou $n \equiv 2$ mod. 8 et $n$ est somme de deux carrés. En conséquence, pour ces valeurs de $n$ , toute extension centrale de $A_{n}$ est groupe de Galois sur $Q$ .

The fields defined by the polynomials constructed in E. Nart and the author in J. Number Theory 16, (1983), 6–13, Th. 2.1, with absolute Galois group the alternating group $A_{n}$ , can be embedded in any central extension of $A_{n}$ if and only if $n \equiv 0 (m o d 8)$ , or $n \equiv 2 (m o d 8)$ and $n$ is a sum of two squares. Consequently, for theses values of $n$ , every central extension of $A_{n}$ occurs as a Galois group over $Q$ .

MR Zbl

DOI : 10.5802/aif.1010

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     author = {Vila, Nuria},
     title = {Polynomials over $Q$ solving an embedding problem},
     journal = {Annales de l'Institut Fourier},
     pages = {79--82},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {35},
     number = {2},
     year = {1985},
     doi = {10.5802/aif.1010},
     mrnumber = {86h:11100},
     zbl = {0546.12006},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1010/}
}

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Vila, Nuria. Polynomials over $Q$ solving an embedding problem. Annales de l'Institut Fourier, Tome 35 (1985) no. 2, pp. 79-82. doi : 10.5802/aif.1010. http://www.numdam.org/articles/10.5802/aif.1010/

Bibliographie
Cité par

[1] B. Huppert, Endliche Gruppen I, Die Grund. der Math. Wiss., 134, Springer, 1967. | MR | Zbl

[2] E. Nart and N. Vila, Equations with absolute Galois group isomorphic to An, J. Number Th., 16 (1983), 6-13. | MR | Zbl

[3] I. Schur, Ùber die Darstellungen der symmetrischen und alternierender Gruppen durch gebrochene lineare Substitutionen, J. reine angew. Math., 139 (1911), 155-250. | JFM

[4] J.-P. Serre, L'invariant de Witt de la forme Tr (x2), Com. Math. Helv., to appear. | Zbl

[5] N. Vila, On central extensions of An as Galois group over Q, to appear. | Zbl

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