Réalisations galoisiennes explicites de certaines familles de $2$-groupes

Behajaina, Angelot

Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 2, pp. 605-630.

Dans cet article, nous construisons, pour certains $2$-groupes $G$, des extensions galoisiennes explicites $E/\mathbb{Q}\left(T\right)$ de groupe $G$ vérifiant $E\cap \overline{\mathbb{Q}}=\mathbb{Q}$. Nous fournissons aussi des progressions arithmétiques explicites d’entiers $t_0$ telles que la spécialisation $E_{t_0}/\mathbb{Q}$ de $E/\mathbb{Q}\left(T\right)$ en $t_0$ soit de groupe $G$.

In this paper, we construct, for some $2$-groups $G$, explicit Galois extensions $E/\mathbb{Q}\left(T\right)$ of group $G$ with $E\cap \overline{\mathbb{Q}}=\mathbb{Q}$. We also provide explicit arithmetic progressions of integers $t_0$ such that the specialization $E_{t_0}/\mathbb{Q}$ of $E/\mathbb{Q}\left(T\right)$ at $t_0$ has Galois group $G$.

Classification : 12E30, 12F12
Mots clés : Théorie inverse de Galois, spécialisation, réalisations explicites, quaternions généralisés

[1] Beckmann, Sybilla On extensions of number fields obtained by specializing branched coverings, J. Reine Angew. Math., Volume 419 (1991), pp. 27-53 | MR | Zbl

[2] Dèbes, Pierre; Ghazi, Nour Galois covers and the Hilbert-Grunwald property, Ann. Inst. Fourier, Volume 62 (2012) no. 3, pp. 989-1013 | DOI | Numdam | MR | Zbl

[3] Dèbes, Pierre; Legrand, François Specialization results in Galois theory, Trans. Am. Math. Soc., Volume 365 (2013) no. 10, pp. 5259-5275 | DOI | MR | Zbl

[4] Fried, Michael D. On Hilbert’s irreducibility theorem, J. Number Theory, Volume 6 (1974), pp. 211-231 | DOI | MR | Zbl

[5] Fried, Michael D. Fields of definition of function fields and Hurwitz families-groups as Galois groups, Commun. Algebra, Volume 5 (1977) no. 1, pp. 17-82 | DOI | MR | Zbl

[6] Fried, Michael D.; Jarden, Moshe Field Arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 11, Springer, 2008 | MR | Zbl

[7] Jensen, Christian U.; Ledet, Arne; Yui, Noriko Generic polynomials. Constructive Aspects of the Inverse Galois Problem, Mathematical Sciences Research Institute Publications, 45, Cambridge University Press, 2002, x+258 pages | Zbl

[8] Legrand, François Specialization results and ramification conditions, Isr. J. Math., Volume 214 (2016) no. 2, pp. 621-650 | DOI | MR | Zbl

[9] Malle, Gunter; Matzat, Bernd Heinrich Inverse Galois theory, Springer Monographs in Mathematics, Springer, 2018, xvii+532 pages | Zbl

[10] Martinais, Dominique; Schneps, Leila Polynômes à groupe de galois diédral, Sémin. Théor. Nombres Bordx., Sér. II, Volume 4 (1992) no. 1, pp. 141-153 | DOI | Numdam | Zbl

[11] Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay Cohomology of number fields, Grundlehren der Mathematischen Wissenschaften, 323, Springer, 2008, xvi+825 pages | MR | Zbl

[12] Schinzel, Andrzej Polynomials with special regard to reducibility, Encyclopedia of Mathematics and Its Applications, 77, Cambridge University Press, 2000, x+558 pages | MR | Zbl

[13] Serre, Jean-Pierre Topics in Galois Theory, Research Notes in Mathematics, 1, Jones and Bartlett Publishers, 1992, xvi+117 pages | MR | Zbl

[14] Stichtenoth, Henning Algebraic function fields and codes, Graduate Texts in Mathematics, 254, Springer, 2009, xiv+355 pages | MR | Zbl

[15] Völklein, Helmut Groups as Galois groups, Cambridge Studies in Advanced Mathematics, 53, Cambridge University Press, 1996, xviii+248 pages | Zbl

[16] Weintraub, Steven Galois Theory, Universitext, Springer, 2009 | Zbl