En utilisant le principe selon lequel le polynôme caractéristique de matrices obtenues comme éléments d’un groupe réductif sur a typiquement un corps de décomposition dont le groupe de Galois est le groupe de Weyl de , nous construisons un polynôme unitaire explicite de degré , à coefficients entiers, dont le corps de décomposition a pour groupe de Galois le groupe de Weyl du groupe exceptionnel de type .
Using the principle that characteristic polynomials of matrices obtained from elements of a reductive group over typically have splitting field with Galois group isomorphic to the Weyl group of , we construct an explicit monic integral polynomial of degree whose splitting field has Galois group the Weyl group of the exceptional group of type .
Publié le :
DOI : https://doi.org/10.5802/jtnb.649
Mots clés : Inverse Galois problem, Weyl group, exceptional algebraic group, random walk on finite group, characteristic polynomial
@article{JTNB_2008__20_3_761_0,
author = {Florent Jouve and Emmanuel Kowalski and David Zywina},
title = {An explicit integral polynomial whose splitting field has {Galois} group $W(\mathbf{E}_8)$},
journal = {Journal de Th\'eorie des Nombres de Bordeaux},
pages = {761--782},
publisher = {Universit\'e Bordeaux 1},
volume = {20},
number = {3},
year = {2008},
doi = {10.5802/jtnb.649},
zbl = {pre05572700},
mrnumber = {2523316},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.649/}
}
Florent Jouve; Emmanuel Kowalski; David Zywina. An explicit integral polynomial whose splitting field has Galois group $W(\mathbf{E}_8)$. Journal de Théorie des Nombres de Bordeaux, Tome 20 (2008) no. 3, pp. 761-782. doi : 10.5802/jtnb.649. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.649/
[1] J. F. Adams, Lectures on exceptional Lie groups. Chicago Lectures in Math., Univ. Chicago Press, 1996. | MR 1428422 | Zbl 0866.22008
[2] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker and R.A. Wilson, Atlas of finite groups; Maximal subgroups and ordinary characters for simple groups, with computational assistance from J. G. Thackray, Oxford University Press, 1985. | MR 827219 | Zbl 0568.20001
[3] N. Berry, A. Dubickas, N. Elkies, B. Poonen and C. J. Smyth, The conjugate dimension of algebraic numbers. Quart. J. Math. 55 (2004), 237–252. | MR 2082091 | Zbl 1062.11064
[4] A. Borel, Linear algebraic groups, 2nd edition. GTM 126, Springer 1991. | MR 1102012 | Zbl 0726.20030
[5] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system, I. The user language. J. Symbolic Comput., 24 (1997), 235–265; also | MR 1484478 | Zbl 0898.68039
[6] N. Bourbaki, Groupes et algèbres de Lie, Chapitres 4, 5, 6. Hermann, 1968. | MR 240238 | Zbl 0483.22001
[7] N. Bourbaki, Groupes et algèbres de Lie, Chapitres 7, 8. Hermann, 1975. | MR 453824
[8] É. Cartan, Sur la réduction à sa forme canonique de la structure d’un groupe de transformations fini et continu. Amer J. Math. 18 (1896), 1–46 (=Oeuvres Complètes, t. I, 293–353).
[9] R.W. Carter, Conjugacy classes in the Weyl group. Compositio Math. 25 (1972), 1–59. | Numdam | MR 318337 | Zbl 0254.17005
[10] C. Chevalley, Sur certains groupes simples. Tôhoku Math. J. 7 (1955), 14–66. | MR 73602 | Zbl 0066.01503
[11] A. Cohen, S. Murray and D.E. Taylor, Computing in groups of Lie type. Math. Comp. 73, Number 247, 1477–1498. | MR 2047097 | Zbl 1062.20049
[12] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4.9, 2007,
[13] E. Kowalski, The large sieve and its applications: arithmetic geometry, random walks and discrete groups. Cambridge Tracts in Math. 175, Cambridge Univ. Press, 2008. | MR 2426239 | Zbl 1177.11080
[14] G. Malle and B.H. Matzat, Inverse Galois theory. Springer Monographs in Math., 1999. | MR 1711577 | Zbl 0940.12001
[15] Y.I Manin, Cubic forms: algebra, geometry, arithmetic. North Holland Math. Library 4, 2nd ed., 1988. | MR 833513 | Zbl 0582.14010
[16] Ya. N. Nuzhin, Weyl groups as Galois groups of a regular extension of the field , (Russian). Algebra i Logika 34 (1995), no. 3, 311–315, 364; translation in Algebra and Logic 34 (1995), no. 3, 169–172. | EuDML 187727 | MR 1364468 | Zbl 0872.12001
[17] PARI/GP, version 2.4.2, Bordeaux, 2007, .
[18] L. Saloff-Coste, Random walks on finite groups. In “Probability on discrete structures”, 263–346, Encyclopaedia Math. Sci., 110, Springer 2004. | MR 2023654 | Zbl 1049.60006
[19] J-P. Serre, Cours d’arithmétique. PUF 1988. | Zbl 0225.12002
[20] T. Shioda, Theory of Mordell-Weil lattices. In Proceedings of ICM 1990 (Kyoto), Vol. I (473–489), Springer, 1991. | MR 1159235 | Zbl 0746.14009
[21] T.A. Springer, Linear algebraic groups, 2nd edition, Progr. Math. 9, Birkhaüser 1998. | MR 1642713 | Zbl 0927.20024
[22] T.A. Springer, Regular elements of finite reflection groups. Invent. math. 25 (1974), 159–198. | EuDML 142286 | MR 354894 | Zbl 0287.20043
[23] R. Steinberg, Lectures on Chevalley groups. Yale Univ. Lecture Notes, 1967. | MR 466335
[24] A. Várilly-Alvarado and D. Zywina, Arithmetic lattices with maximal Galois action. To appear in LMS J. Comput. Math. | Zbl 1252.11055
[25] V.E. Voskresenskii, Maximal tori without effect in semisimple algebraic groups (Russian). Matematicheskie Zametki, Vol. 44 (1988), 309–318; English translation: Mathematical Notes 44, 651–655. | MR 972194 | Zbl 0699.20037