Splitting fields for $E_{8}$-torsors
Burt Totaro
Source: Duke Math. J. Volume 121, Number 3 (2004), 425-455.
Abstract
We show that every algebraic group of type $E_{8}$ over any field becomes split over some field extension of degree dividing $2^{6}\cdot 3^{2}\cdot 5=2880$. This improves a bound by Tits and, in fact, is optimal.
Primary Subjects: 11E72
Secondary Subjects: 14M17, 20G15
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Alternatively, the document is available for a cost of $25. Select the "buy article" button below to purchase this document from a secured VeriSign, Inc. site.
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.dmj/1078170755
Digital Object Identifier: doi:10.1215/S0012-7094-04-12132-3
Mathematical Reviews number (MathSciNet):
MR2040282
Zentralblatt MATH identifier:
1048.11031
References
[1] M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), 469--514.
Mathematical Reviews (MathSciNet):
MR0746539
[2] R. Baeza, Quadratic Forms over Semilocal Rings, Lecture Notes in Math. 655, Springer, Berlin, 1978.
Mathematical Reviews (MathSciNet):
MR0491773
Zentralblatt MATH:
0382.10014
[3] E. Bayer-Fluckiger and H. W. Lenstra Jr., Forms in odd degree extensions and self-dual normal bases, Amer. J. Math. 112 (1990), 359--373.
Mathematical Reviews (MathSciNet):
MR1055648
[4] F. van der Blij and T. A. Springer, The arithmetics of octaves and of the group $G_2$, Indag. Math. 21 (1959), 406--418.
Mathematical Reviews (MathSciNet):
MR0152555
[5] A. Borel and T. A. Springer, Rationality properties of linear algebraic groups, II, Tôhoku Math. J. (2) 20 (1968), 443--497.
Mathematical Reviews (MathSciNet):
MR0244259
[6] N. Bourbaki, Éléments de mathématique, fasc. 34: Groupes et algèbres de Lie, chapitres 4--6, Masson, Paris, 1981.
Mathematical Reviews (MathSciNet):
MR0240238
[7] R. Brauer, On groups whose order contains a prime number to the first power, I, II, Amer. J. Math. 64 (1942), 401--420., 421--440. ,
[8] —, Über endliche lineare Gruppen von Primzahlgrad, Math. Ann. 169 (1967), 73--96.
Mathematical Reviews (MathSciNet):
MR0206088
[9] —, "On simple groups of order $5\cdot 3^a\cdot 2^b$" in Collected Papers, Vol. 2: Finite Groups, ed. P. Fong and W. J. Wong, Mathematicians of Our Time 18, MIT Press, Cambridge, Mass., 1980, 421--470.
Mathematical Reviews (MathSciNet):
MR0581122
Zentralblatt MATH:
0438.01008
[10] W. Burnside, Theory of Groups of Finite Order, 2nd ed., Dover, New York, 1955.
Mathematical Reviews (MathSciNet):
MR0069818
Zentralblatt MATH:
0064.25105
[11] R. Carter, Conjugacy classes in the Weyl group, Compositio Math. 25 (1972), 1--59.
Mathematical Reviews (MathSciNet):
MR0318337
[12] F. Cole, Simple groups as far as order 660, Amer. J. Math. 15 (1893), 303--315.
[13] J.-L. Colliot-Thélène and D. Coray, L'équivalence rationnelle sur les points fermés des surfaces rationnelles fibrées en coniques, Compositio Math. 39 (1979), 301--332.
Mathematical Reviews (MathSciNet):
MR0550646
[14] 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, Oxford Univ. Press, New York, 1985.
Mathematical Reviews (MathSciNet):
MR0827219
Zentralblatt MATH:
0568.20001
[15] W. Feit, "On finite linear groups in dimension at most 10" in Proceedings of the Conference on Finite Groups (Park City, Utah, 1975), ed. W. R. Scott and F. Gross, Academic Press, New York, 1976, 397--407.
Mathematical Reviews (MathSciNet):
MR0412294
Zentralblatt MATH:
0374.20009
[16] The GAP Group, GAP -- Groups, Algorithms, and Programming, version 4.2, Lehrstuhl D für Mathematik, Rheinisch-Westfälische Technische Hochschule Aachen, Aachen, Germany; University of St. Andrews, Fife, Scotland, 2000, http://gap.dcs.st-and.ac.uk/gap
[17] R. S. Garibaldi, The Rost invariant has trivial kernel for quasi-split groups of low rank, Comment. Math. Helv. 76 (2001), 684--711.
Mathematical Reviews (MathSciNet):
MR1881703
Digital Object Identifier: doi:10.1007/s00014-001-8325-8
[18] P. Gille, La $R$-équivalence sur les groupes algébriques réductifs définis sur un corps global, Inst. Hautes Études Sci. Publ. Math. 86 (1997), 199--235.
Mathematical Reviews (MathSciNet):
MR1608570
[19] —, Invariants cohomologiques de Rost en caractéristique positive, $K$-Theory 21 (2000), 57--100.
Mathematical Reviews (MathSciNet):
MR1802626
Digital Object Identifier: doi:10.1023/A:1007839108933
[20] D. Gorenstein, Finite Groups, 2nd ed., Chelsea, New York, 1980.
Mathematical Reviews (MathSciNet):
MR0569209
[21] R. L. Griess Jr., Elementary abelian $p$-subgroups of algebraic groups, Geom. Dedicata 39 (1991), 253--305.
Mathematical Reviews (MathSciNet):
MR1123145
Digital Object Identifier: doi:10.1007/BF00150757
[22] A. Grothendieck, "Torsion homologique et sections rationnelles" in Anneaux de Chow et applications, Séminaire C. Chevalley 2, Secrétariat mathématique, Paris, 1958, exp. 5.
Mathematical Reviews (MathSciNet):
MR0110704
[23] N. Jacobson, Composition algebras and their automorphisms, Rend. Circ. Mat. Palermo (2) 7 (1958), 55--80.
Mathematical Reviews (MathSciNet):
MR0101253
[24] P. Kleidman and M. Liebeck, The Subgroup Structure of the Finite Classical Groups, London Math. Soc. Lecture Note Ser. 129, Cambridge Univ. Press, Cambridge, 1990.
Mathematical Reviews (MathSciNet):
MR1057341
Zentralblatt MATH:
0697.20004
[25] M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol, The Book of Involutions, Amer. Math. Soc. Colloq. Publ. 44, Amer. Math. Soc., Providence, 1998.
Mathematical Reviews (MathSciNet):
MR1632779
Zentralblatt MATH:
0955.16001
[26] D. H. Lehmer, On a problem of Störmer, Illinois J. Math. 8 (1964), 57--79.
Mathematical Reviews (MathSciNet):
MR0158849
[27] H. P. Petersson and M. L. Racine, An elementary approach to the Serre-Rost invariant of Albert algebras, Indag. Math. (N.S.) 7 (1996), 343--365.
Mathematical Reviews (MathSciNet):
MR1621373
[28] A. Pfister, Quadratische Formen in beliebigen Körpern, Invent. Math. 1 (1966), 116--132.
Mathematical Reviews (MathSciNet):
MR0200270
[29] M. Rost, A ($\modd 3$) invariant for exceptional Jordan algebras, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), 823--827.
Mathematical Reviews (MathSciNet):
MR1138557
[30] L. H. Rowen, Ring Theory, Vol. II, Pure Appl. Math. 128, Academic Press, Boston, 1988.
Mathematical Reviews (MathSciNet):
MR0945718
[31] —, Are $p$-algebras having cyclic quadratic extensions necessarily cyclic?, J. Algebra 215 (1999), 205--228.
Mathematical Reviews (MathSciNet):
MR1684174
Digital Object Identifier: doi:10.1006/jabr.1998.7734
[32] J.-J. Sansuc, Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. Reine Angew. Math. 327 (1981), 12--80.
Mathematical Reviews (MathSciNet):
MR0631309
[33] I. Schur, "Über eine Klasse von endlichen Gruppen linearer Substitutionen" in Sitz. der Preussischen Akad. Berlin (1905), 77--91.; Gesammelte Abhandlungen, Vol. 1, Springer, Berlin, 1973, 128--142.
Mathematical Reviews (MathSciNet):
MR0462891
Zentralblatt MATH:
0274.01054
[34] J.-P. Serre, Lectures on the Mordell-Weil Theorem, Aspects Math. E15, Vieweg, Braunhschweig, Germany, 1989.
Mathematical Reviews (MathSciNet):
MR1002324
Zentralblatt MATH:
0676.14005
[35] —, Cohomologie galoisienne, 5th ed., Lecture Notes in Math. 5, Springer, Berlin, 1994.
Mathematical Reviews (MathSciNet):
MR1324577
Zentralblatt MATH:
0812.12002
[36] —, Cohomologie galoisienne: progrès et problèmes, Astérisque 227 (1995), 229--257., Séminaire Bourbaki 1993/94, no. 783.
Mathematical Reviews (MathSciNet):
MR1321649
[37] C. C. Sims, "Computational methods in the study of permutation groups" in Computational Problems in Abstract Algebra (Oxford, 1967), ed. J. Leech, Pergamon, Oxford, 1970, 169--183.
Mathematical Reviews (MathSciNet):
MR0257203
Zentralblatt MATH:
0215.10002
[38] T. A. Springer, Sur les formes quadratiques d'indice zéro, C. R. Acad. Sci. Paris Sér. I Math. 234 (1952), 1517--1519.
Mathematical Reviews (MathSciNet):
MR0047021
[39] D. A. Suprunenko, Soluble and Nilpotent Linear Groups, Amer. Math. Soc., Providence, 1963.
Mathematical Reviews (MathSciNet):
MR0154918
[40] J. Tits, Sur les degrés des extensions de corps déployant les groupes algébriques simples, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), 1131--1138.
Mathematical Reviews (MathSciNet):
MR1194504
[41] B. Totaro, The torsion index of $E_8$ and other groups, preprint, 2003.
[42] B. Veisfeiler [B. Ju. Veĭsfeĭler], Certain properties of singular semisimple algebraic groups over non-closed fields, Trudy Moskov. Mat. Obshch. 20 (1969), 111--136.; English translation in Trans. Moscow Math. Soc. (1971), 109--134.
Mathematical Reviews (MathSciNet):
MR0257091
[43] W. C. Waterhouse, Introduction to Affine Group Schemes, Grad. Texts in Math. 66, Springer, New York, 1979.
Mathematical Reviews (MathSciNet):
MR0547117
Zentralblatt MATH:
0442.14017
Duke Mathematical Journal
