An explicit integral polynomial whose splitting field has Galois group W(E 8 )
Florent Jouve1 ; Emmanuel Kowalski2 ; David Zywina3
1 Dept. of Mathematics, The University of Texas at Austin 1 University Station C1200 Austin, TX, 78712, USA.
2 ETH Zürich – DMATH Rämistrasse 101 8092 Zürich, Switzerland
3 Department of Mathematics, University of Pennsylvania Philadelphia, PA 19104-6395, USA
Journal de théorie des nombres de Bordeaux, Tome 20 (2008) no. 3, pp. 761-782

Using the principle that characteristic polynomials of matrices obtained from elements of a reductive group G over Q typically have splitting field with Galois group isomorphic to the Weyl group of G, we construct an explicit monic integral polynomial of degree 240 whose splitting field has Galois group the Weyl group of the exceptional group of type E 8 .

En utilisant le principe selon lequel le polynôme caractéristique de matrices obtenues comme éléments d’un groupe réductif G sur Q a typiquement un corps de décomposition dont le groupe de Galois est le groupe de Weyl de G, nous construisons un polynôme unitaire explicite de degré 240, à coefficients entiers, dont le corps de décomposition a pour groupe de Galois le groupe de Weyl du groupe exceptionnel de type E 8 .

DOI : 10.5802/jtnb.649
Keywords: Inverse Galois problem, Weyl group, exceptional algebraic group, random walk on finite group, characteristic polynomial

Florent Jouve 1 ; Emmanuel Kowalski 2 ; David Zywina 3

1 Dept. of Mathematics, The University of Texas at Austin 1 University Station C1200 Austin, TX, 78712, USA.
2 ETH Zürich – DMATH Rämistrasse 101 8092 Zürich, Switzerland
3 Department of Mathematics, University of Pennsylvania Philadelphia, PA 19104-6395, USA 
@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},
     year = {2008},
     publisher = {Universit\'e Bordeaux 1},
     volume = {20},
     number = {3},
     doi = {10.5802/jtnb.649},
     mrnumber = {2523316},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.649/}
}
TY  - JOUR
AU  - Florent Jouve
AU  - Emmanuel Kowalski
AU  - David Zywina
TI  - An explicit integral polynomial whose splitting field has Galois group $W(\mathbf{E}_8)$
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2008
SP  - 761
EP  - 782
VL  - 20
IS  - 3
PB  - Université Bordeaux 1
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.649/
DO  - 10.5802/jtnb.649
LA  - en
ID  - JTNB_2008__20_3_761_0
ER  - 
%0 Journal Article
%A Florent Jouve
%A Emmanuel Kowalski
%A David Zywina
%T An explicit integral polynomial whose splitting field has Galois group $W(\mathbf{E}_8)$
%J Journal de théorie des nombres de Bordeaux
%D 2008
%P 761-782
%V 20
%N 3
%I Université Bordeaux 1
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.649/
%R 10.5802/jtnb.649
%G en
%F JTNB_2008__20_3_761_0
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

[1] J. F. Adams, Lectures on exceptional Lie groups. Chicago Lectures in Math., Univ. Chicago Press, 1996. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[4] A. Borel, Linear algebraic groups, 2nd edition. GTM 126, Springer 1991. | Zbl | MR

[5] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system, I. The user language. J. Symbolic Comput., 24 (1997), 235–265; also http://magma.maths.usyd.edu.au/magma/ | Zbl | MR

[6] N. Bourbaki, Groupes et algèbres de Lie, Chapitres 4, 5, 6. Hermann, 1968. | Zbl | MR

[7] N. Bourbaki, Groupes et algèbres de Lie, Chapitres 7, 8. Hermann, 1975. | MR

[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 1 , 293–353).

[9] R.W. Carter, Conjugacy classes in the Weyl group. Compositio Math. 25 (1972), 1–59. | Zbl | MR | Numdam

[10] C. Chevalley, Sur certains groupes simples. Tôhoku Math. J. 7 (1955), 14–66. | Zbl | MR

[11] A. Cohen, S. Murray and D.E. Taylor, Computing in groups of Lie type. Math. Comp. 73, Number 247, 1477–1498. | Zbl | MR

[12] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4.9, 2007, www.gap-system.org

[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 | Zbl

[14] G. Malle and B.H. Matzat, Inverse Galois theory. Springer Monographs in Math., 1999. | Zbl | MR

[15] Y.I Manin, Cubic forms: algebra, geometry, arithmetic. North Holland Math. Library 4, 2nd ed., 1988. | Zbl | MR

[16] Ya. N. Nuzhin, Weyl groups as Galois groups of a regular extension of the field Q, (Russian). Algebra i Logika 34 (1995), no. 3, 311–315, 364; translation in Algebra and Logic 34 (1995), no. 3, 169–172. | Zbl | MR | EuDML

[17] PARI/GP, version 2.4.2, Bordeaux, 2007, http://pari.math.u-bordeaux.fr/.

[18] L. Saloff-Coste, Random walks on finite groups. In “Probability on discrete structures”, 263–346, Encyclopaedia Math. Sci., 110, Springer 2004. | Zbl | MR

[19] J-P. Serre, Cours d’arithmétique. PUF 1988. | Zbl

[20] T. Shioda, Theory of Mordell-Weil lattices. In Proceedings of ICM 1990 (Kyoto), Vol. I (473–489), Springer, 1991. | Zbl | MR

[21] T.A. Springer, Linear algebraic groups, 2nd edition, Progr. Math. 9, Birkhaüser 1998. | Zbl | MR

[22] T.A. Springer, Regular elements of finite reflection groups. Invent. math. 25 (1974), 159–198. | Zbl | MR | EuDML

[23] R. Steinberg, Lectures on Chevalley groups. Yale Univ. Lecture Notes, 1967. | MR

[24] A. Várilly-Alvarado and D. Zywina, Arithmetic E 8 lattices with maximal Galois action. To appear in LMS J. Comput. Math. | Zbl

[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. | Zbl | MR

Cité par Sources :