Abstract
We provide a survey of past research and a list of open problems regarding central simple algebras and the Brauer group over a field, intended both for experts and for beginners.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. A. Albert, Normal division algebras of degree four over an algebraic field, Trans. Amer. Math. Soc. 34 (1932), no. 2, 363–372.
A. A. Albert, Non-cyclic division algebras of degree and exponent four, Trans. Amer. Math. Soc. 35 (1933), 112–121.
A. A. Albert, Normal division algebras of degree 4 over F of characteristic 2, Amer. J. Math. 56 (1934), nos. 1–4, 75–86.
A. A. Albert, Simple algebras of degree p e over a centrum of characteristic p, Trans. Amer. Math. Soc. 40 (1936), no. 1, 112–126.
[Al38] A. A. Albert, A note on normal division algebras of prime degree, Bull. Amer. Math. Soc. 44 (1938), no. 10, 649–652.
A. A. Albert, Structure of Algebras, Amer. Math. Soc. Colloq. Publ., Vol. 24, Amer. Math. Soc., Providence, RI, 1961, revised printing.
A. A. Albert, Tensor products of quaternion algebras, Proc. Amer. Math. Soc. 35 (1972), 65–66.
E. S. Allman, M. M. Schacher, Division algebras with PSL(2, q)-Galois maximal subfields, J. Algebra 240 (2001), no. 2, 808–821.
S. A. Amitsur, Generic splitting fields of central simple algebras, Ann. of Math. (2) 62 (1955), 8–43.
S. A. Amitsur, On central division algebras, Israel J. Math. 12 (1972), 408–422.
S. A. Amitsur, Division algebras. A survey, in: Algebraists’ Homage: Papers in Ring Theory and Related topics (New Haven, CN, 1981), Contemp. Math., vol. 13, Amer. Math. Soc., Providence, RI, 1982, pp. 3–26.
S. A. Amitsur, Galois splitting fields of a universal division algebra J. Algebra, 143 (1991) 236–245.
S. A. Amitsur, L. H. Rowen, J.-P. Tignol, Division algebras of degree 4 and 8 with involution, Israel J. Math. 33 (1979), 133–148.
S. A. Amitsur, D. J. Saltman, Generic abelian crossed products and p-algebras, J. Algebra 51 (1978), 76–87.
[ArJa] R. Aravire, B. Jacob, Relative Brauer groups in characteristic p, Proc. Amer. Math. Soc. 137 (2009), no. 4, 1265–1273.
M. Artin, Brauer–Severi varieties, in: Brauer Groups in Ring Theory and Algebraic Geometry (Wilrijk, 1981), Lecture Notes in Math., Vol. 917, Springer-Verlag, Berlin, 1982, pp. 194–210.
M. Artin, H. P. F. Swinnerton-Dyer, The Shafarevich–Tate conjecture for pencils of elliptic curves on K3 surfaces, Invent. Math. 20 (1973), 249–266.
S. Baek, Invariants of simple algebras, PhD thesis, UCLA, 2010.
S. Baek, A. Merkurjev, Invariants of simple algebras, Manuscripta Math. 129 (2009), no. 4, 409–421.
S. Baek, A. Merkurjev, Essential dimension of central simple algebras, Acta Math. (to appear).
E. Bayer-Fluckiger, H. W. Lenstra, Jr., Forms in odd degree extensions and self-dual normal bases, Amer. J. Math. 112 (1990), no. 3, 359–373.
E. Bayer-Fluckiger, D. B. Shapiro, J.-P. Tignol, Hyperbolic involutions, Math. Z. 214 (1993), no. 3, 461–476.
K. J. Becher, D. W. Hoffmann, Symbol lengths in Milnor K -theory, Homology, Homotopy Appl. 6 (2004), no. 1, 17–31.
E. Beneish, Induction theorems on the stable rationality of the center of the ring of generic matrices, Trans. Amer. Math. Soc. 350 (1998), no. 9, 3571–3585.
E. Beneish, Centers of generic algebras with involution, J. Algebra 294 (2005), no. 1, 41–50.
G. Berhuy, G. Favi, Essential dimension: a functorial point of view (after A. Merkurjev), Doc. Math. 8 (2003), 279–330.
G. Berhuy, C. Frings, J.-P. Tignol, Galois cohomology of the classical groups over imperfect fields, J. Pure Appl. Algebra 211 (2007), 307–341.
C. Bessenrodt, L. Le Bruyn, Stable rationality of certain PGL n -quotients, Invent. Math. 104 (1991), no. 1, 179–199.
N. Bourbaki, Algebra II, Springer-Verlag, New York, 1988.
N. Bourbaki, Commutative Algebra, Chaps. 1–7, Springer-Verlag, New York, 1988.
R. Brauer, On normal division algebras of index 5, Proc. Natl. Acad. Sci. USA 24 (1938), no. 6, 243–246.
R. Brauer, E. Noether, Über minimale Zerfällungskörper irreduzibler Darstellungen, Sitz. Akad. Berlin (1927), 221–228.
P. Brosnan, Z. Reichstein, A. Vistoli, Essential dimension, spinor groups, and quadratic forms, Ann. of Math. (2) 171 (2010), no. 1, 533–544.
A. Brumer, M. Rosen, On the size of the Brauer group, Proc. Amer. Math. Soc. 19 (1968), 707–711.
E. S. Brussel, Noncrossed products and nonabelian crossed products over ℚ(t) and ℚ((t)), Amer. J. Math. 117 (1995), no. 2, 377–393.
E. S. Brussel, Division algebras not embeddable in crossed products, J. Algebra 179 (1996), no. 2, 631–655.
E. S. Brussel, An arithmetic obstruction to division algebra decomposability, Proc. Amer. Math. Soc. 128 (2000), no. 8, 2281–2285.
E. S. Brussel, The division algebras and Brauer group of a strictly Henselian field, J. Algebra 239 (2001), no. 1, 391–411.
E. S. Brussel, On Saltman’s p-adic curves papers, in: Quadratic Forms, Linear Algebraic Groups, and Cohomology (J.-L. Colliot-Thélène, S. Garibaldi, R. Sujatha, V. Suresh, eds.), Dev. Math., Vol. 18, Springer, New York, 2010, pp. 13–38.
E. Brussel, K. McKinnie, E. Tengan, Indecomposable and noncrossed product division algebras over function fields of smooth p-adic curves, arxiv:0907.0670, July 2009.
E. Brussel, E. Tengan, Division algebras of prime period over function fields of p-adic curves (tame case ), arxiv.org:1003.1906, 2010.
J. Buhler, Z. Reichstein, On the essential dimension of a finite group, Compos. Math. 106 (1997), no. 2, 159–179.
F. Châtelet, Variations sur un thème de H. Poincaré, Ann. Sci. École Norm. Sup. 61 (1944), no. 3, 249–300.
D. Chillag, J. Sonn, Sylow-metacyclic groups and ℚ-admissibility, Israel J. Math. 40 (1981), nos. 3–4, 307–323.
J.-L. Colliot-Thélène, Die Brauersche Gruppe; ihre Verallgemeinerungen und Anwendungen in der Arithmetischen Geometrie, Brauer Tagung (Stuttgart, March 2001), http://www.math.u-psud.fr/~colliot/brauer.ps.
J.-L. Colliot-Thélène, S. Garibaldi, R. Sujatha, V. Suresh (eds.), Quadratic Forms, Linear Algebraic Groups, and Cohomology, Dev. Math., Vol. 18, Springer, New York, 2010.
A. J. de Jong, The period-index problem for the Brauer group of an algebraic surface, Duke Math. J. 123 (2004), no. 1, 71–94.
P. K. Draxl, Skew Fields, London Math. Soc. Lecture Note Ser., Vol. 81, Cambridge University Press, Cambridge, 1983.
I. Efrat, On fields with finite Brauer groups, Pacific J. Math. 177 (1997), no. 1, 33–46.
R. S. Elman, N. Karpenko, A. Merkurjev, The Algebraic and Geometric Theory of Quadratic Forms, Amer. Math. Soc., Providence, RI, 2008.
G. Favi, M. Florence, Tori and essential dimension, J. Algebra 319 (2008), no. 9, 3885–3900.
B. Fein, M. M. Schacher, Brauer groups of fields algebraic over ℚ, J. Algebra 43 (1976), no. 1, 328–337.
B. Fein, D. J. Saltman, M. M. Schacher, Crossed products over rational function fields, J. Algebra 156 (1993), no. 2, 454–493.
P. Feit, W. Feit, The K -admissibility of SL(2; 5), Geom. Dedicata 36 (1990), no. 1, 1–13.
W. Feit, The ℚ-admissibility of 2A 6 and 2A 7, in: Algebraic Geometry and its Applications (West Lafayette, IN, 1990), Springer, New York, 1994, pp. 197–202.
W. Feit, SL(2; 11) is ℚ-admissible, J. Algebra 257 (2002), no. 2, 244–248.
W. Feit, PSL2(11) is admissible for all number fields, in: Algebra, Arithmetic and Geometry with Applications (West Lafayette, IN, 2000), Springer, Berlin, 2004, pp. 295–299.
W. Feit, P. Vojta, Examples of some ℚ-admissible groups, J. Number Theory 26 (1987), no. 2, 210–226.
T. J. Ford, Every finite abelian group is the Brauer group of a ring, Proc. Amer. Math. Soc. 82 (1981), no. 3, 315–321.
E. Formanek, The ring of generic matrices, J. Algebra 258 (2002), 310–320.
S. Garibaldi, Cohomological Invariants: Exceptional Groups and Spin Groups, with an appendix by Detlev W. Hoffmann, Mem. Amer. Math. Soc., Vol. 200, no. 937, Amer. Math. Soc., Providence, RI, 2009.
S. Garibaldi, Orthogonal involutions on algebras of degree 16 and the Killing form of E 8, with an appendix by K. Zainoulline, in: Quadratic Forms–Algebra, Arithmetic, and Geometry (R. Baeza, W.K. Chan, D.W. Hoffmann, R. Schulze-Pillot, eds.), Contemp. Math., Vol. 493, Amer. Math. Soc, Providence, RI, 2009, pp. 131–162.
S. Garibaldi, A. Merkurjev, J-P. Serre, Cohomological invariants in Galois cohomology, Univ. Lecture Ser., Vol. 28, Amer. Math. Soc., Providence, RI, 2003.
S. Garibaldi, D. J. Saltman, Quaternion algebras with the same subfields, in: Quadratic Forms, Linear Algebraic Groups, and Cohomology (J.-L. Colliot-Thélène, S. Garibaldi, R. Sujatha, V. Suresh, eds.), Dev. Math., Vol. 18, Springer, New York, 2010, pp. 217–230.
P. Gille, Le problème de Kneser–Tits, Astérisque 326 (2009), 39–82.
P. Gille, T. Szamuely, Central Simple Algebras and Galois Cohomology, Cambridge Stud. Adv. Math., Vol. 101, Cambridge University Press, Cambridge, 2006.
L. I. Gordon, Normal division algebras of degree four, Master’s thesis, University of Chicago, 1940.
B. Gordon, M. M. Schacher, Quartic coverings of a cubic, in: Number Theory and Algebra, Academic Press, New York, 1977, pp. 97–101.
B. Gordon, M. M. Schacher, The admissibility of A 5, J. Number Theory 11 (1979), no. 4, 498–504.
D. E. Haile, On dihedral algebras and conjugate splittings, in: Rings, Extensions, and Cohomology (Evanston, IL, 1993), Lecture Notes Pure Appl. Math., Vol. 159, Dekker, New York, 1994, pp. 107–111.
D. Haile, M.-A. Knus, M. Rost, J.-P. Tignol, Algebras of odd degree with involution, trace forms and dihedral extensions, Israel J. Math. 96 (1996), 299–340.
T. Hanke, A twisted Laurent series ring that is a noncrossed product, Israel J. Math. 150 (2005), 199–203.
D. Harbater, J. Hartmann, D. Krashen, Applications of patching to quadratic forms and central simple algebras, Invent. Math. 178 (2009), 231–263.
D. Harbater, J. Hartmann, D. Krashen, Patching subfields of division algebras, Trans. Amer. Math. Soc. (to appear).
G. Harder, R. P. Langlands, M. Rapoport, Algebraische Zyklen auf Hilbert–Blumenthal Flächen, J. Reine Angew. Math. 366 (1986), 53–120.
H. Hasse, Existenz gewisser algebraischer Zahlkörper, Sitz. Akad. Berlin (1927), 229–234.
I. N. Herstein, Noncommutative Rings, Carus Math. Monogr. Vol. 15, Math. Assoc. Amer., Washington, DC, 1968.
M. Hindry, J. H. Silverman, Diophantine Geometry, Graduate Texts in Mathematics, Vol. 201, Springer-Verlag, New York, 2000.
L. Illusie, Complexe de Rham–Witt et cohomologie cristalline, Ann. Sci. École Norm. Sup. 12 (1979), no. 4, 501–661.
B. Jacob, Indecomposable division algebras of prime exponent, J. Reine Angew. Math. 413 (1991), 181–197.
B. Jacob, A. Wadsworth, A new construction of noncrossed product algebras, Trans. Amer. Math. Soc. 293 (1986), no. 2, 693–721.
B. Jacob, A. Wadsworth, Division algebras with no common subfields, Israel J. Math. 83 (1993), no. 3, 353–360.
N. Jacobson, Finite-Dimensional Division Algebras over Fields, Springer-Verlag, Berlin, 1996.
C. U. Jensen, A. Ledet, N. Yui, Generic Polynomials. Constructive Aspects of the Inverse Galois Problem, Math. Sci. Res. Inst. Publ., Vol. 45, Cambridge University Press, Cambridge, 2002.
B. Kahn, Quelques remarques sur le u-invariant, Sémin. Théor. Nombres Bordeaux (2) 2 (1990), 155–161; erratum: (2) 3 (1991), 247.
B. Kahn, Comparison of some field invariants, J. Algebra 232 (2000), no. 2, 485–492.
B. Kahn, Formes Quadratiques sur un Corps, Soc. Math. France, Paris, 2008.
N. A. Karpenko, Torsion in CH2 of Severi–Brauer varieties and indecomposability of generic algebras, Manuscripta Math. 88 (1995), no. 1, 109–117.
N. A. Karpenko, Codimension 2 cycles on Severi–Brauer varieties, K-Theory 13 (1998), no. 4, 305–330.
N. A. Karpenko, Three theorems on common splitting fields of central simple algebras, Israel J. Math. 111 (1999), 125–141.
N. A. Karpenko, On anisotropy of orthogonal involutions, J. Ramanujan Math. Soc. 15 (2000), no. 1, 1–22.
N. A. Karpenko, Isotropy of orthogonal involutions, November 2009, http://www.math.uni-bielefeld.de/LAG/.
N. A. Karpenko, On isotropy of quadratic pair, in: Quadratic Forms–Algebra, Arithmetic, and Geometry (R. Baeza, W.K. Chan, D.W. Hoffmann, and R. Schulze-Pillot, eds.), Contemp. Math., Vol. 493, Amer. Math. Soc., Providence, RI, 2009, pp. 211–217.
N. A. Karpenko, Canonical dimension, in Proceedings of the International Congress of Mathematicians 2010, World Scientific, Singapore, 2010.
N. A. Karpenko, Hyperbolicity of orthogonal involutions, with an appendix by J.-P. Tignol, Doc. Math., extra vol. Suslin (2010), 371–392.
N. A. Karpenko, A. S. Merkurjev, Essential dimension of finite p-groups, Invent. Math. 172 (2008), no. 3, 491–508.
P. I. Katsylo, Stable rationality of fields of invariants of linear representations of the groups PSL6 and PSL12, Math. Notes 48 (1990), no. 2, 751–753.
I. Kersten, Brauergruppen von Körpern, Vieweg, Braunschweig, 1990.
I. Kersten, U. Rehmann, Generic splitting of reductive groups, Tôhoku Math. J. (2) 46 (1994), 35–70.
H. Kisilevsky, J. Sonn, On the n-torsion subgroup of the Brauer group of a number field, J. Théor Nombres de Bordeaux 15 (2003), 199–203.
H. Kisilevsky, J. Sonn, Abelian extensions of global fields with constant local degrees, Math. Res. Lett. 13 (2006), 599–605.
M.-A. Knus, A. S. Merkurjev, M. Rost, J.-P. Tignol, The Book of Involutions, Amer. Math. Soc., Colloq. Publ., Vol. 44, Amer. Math. Soc., Providence, RI, 1998.
D. Krashen, Severi-Brauer varieties of semidirect product algebras, Doc. Math. 8 (2003), 527–546.
D. Krashen, Corestrictions of algebras and splitting fields, Trans. Amer. Math. Soc. 362 (2010), 4781–4792.
D. Krashen, K. McKinnie, Distinguishing division algebras by finite splitting fields, Manuscripta Math. 134 (2011), 171–182.
A. Laghribi, Isotropie de certaines formes quadratiques de dimensions 7 et 8 sur le corps des fonctions d’une quadrique, Duke Math. J. 85 (1996), no. 2, 397–410.
T. Y. Lam, On the linkage of quaternion algebras, Bull. Belg. Math. Soc. Simon Stevin 9 (2002), no. 3, 415–418.
T. Y. Lam, D. Leep, J.-P. Tignol, Biquaternion algebras and quartic extensions, Inst. Hautes Études Sci. Publ. Math. (1993), no. 77, 63–102.
A. Langer, Zero-cycles on Hilbert–Blumenthal surfaces, Duke Math. J. 103 (2000), no. 1, 131–163.
A. Langer, On the Tate conjecture for Hilbert modular surfaces in finite characteristic, J. Reine Angew. Math. 570 (2004), 219–228.
L. Le Bruyn, Centers of generic division algebras, the rationality problem 1965–1990, Israel J. Math. 76 (1991), 97–111.
A. Ledet, Finite groups of essential dimension one, J. Algebra 311 (2007), no. 1, 31–37.
N. Lemire, Essential dimension of algebraic groups and integral representations of Weyl groups, Transform. Groups 9 (2004), no. 4, 337–379.
S. Liedahl, K -admissibility of wreath products of cyclic p-groups, J. Number Theory 60 (1996), no. 2, 211–232.
M. Lieblich, Twisted sheaves and the period-index problem, Compos. Math. 144 (2008), 1–31.
Q. Liu, D. Lorenzini, M. Raynaud, Néron models, Lie algebras, and reduction of curves of genus one, Invent. Math. 157 (2004), no. 3, 455–518.
Q. Liu, D. Lorenzini, M. Raynaud, On the Brauer group of a surface, Invent. Math. 159 (2005), no. 3, 673–676.
M. Lorenz, Z. Reichstein, L. H. Rowen, D. J. Saltman, Fields of definition for division algebras, J. London Math. Soc. (2) 68 (2003), no. 3, 651–670.
R. Lötscher, M. MacDonald, A. Meyer, Z. Reichstein, Essential p-dimension of algebraic tori, arXiv:0910.5574v1, 2009.
P. Mammone, On the tensor product of division algebras, Arch. Math. (Basel) 58 (1992), no. 1, 34–39.
P. Mammone, A. Merkurjev, On the corestriction of p n -symbol, Israel J. Math. 76 (1991), nos. 1–2, 73–79.
P. Mammone, J.-P. Tignol, Dihedral algebras are cyclic, Proc. Amer. Math. Soc. 101 (1987), no. 2, 217–218.
E. Matzri, All dihedral algebras of degree 5 are cyclic, Proc. Amer. Math. Soc. 136 (2008), 1925–1931.
K. McKinnie, Prime to p extensions of the generic abelian crossed product, J. Algebra 317 (2007), no. 2, 813–832.
K. McKinnie, Indecomposable p-algebras and Galois subfields in generic abelian crossed products, J. Algebra 320 (2008), no. 5, 1887–1907.
А. С. Меркуръев, О символе норменното вычета степени два, ДАН СССР 261 (1981), no. 3, 542–547. Engl. transl.: A. S. Merkurjev, On the norm residue symbol of degree 2, Soviet Math. Dokl. 24 (1981), 546–551.
A. S. Merkurjev, Brauer groups of fields, Comm. Algebra 11 (1983), 2611–2624.
А. С. Меркуръев, Делимые группы Брауэра, УМН 40 (1985), no. 2(242), 213–214. [A. S. Merkurjev, Divisible Brauer groups, Uspekhi Mat. Nauk 40 (1985), no. 2(242), 213–214 (Russian).]
А. С. Меркуръев, О строении группы Брауэра полей, Иэв. АН СССР, сер. мат. 49 (1985), no. 4, 828–846. Engl. transl.: A. S. Merkurjev, On the structure of the Brauer group of fields, Math. USSR-Izv. 27 (1986), 141–157.
А. С. Меркуръев, Простые алгебры и квадратичные формы, Изв. АН СССР, сер. мат. 55 (1991), no. 1, 218–224. Engl. transl.: A. S. Merkurjev, Simple algebras and quadratic forms, Math. USSR-Izv. 38 (1992), no. 1, 215–221.
A. S. Merkurjev, Generic element in SK 1 for simple algebras, K-Theory 7 (1993), 1–3.
A. S. Merkurjev, The group SK 1 for simple algebras, K-Theory 37 (2006), no. 3, 311–319.
A. S. Merkurjev, Essential p-dimension of PGL p 2 , J. Amer. Math. Soc. 23 (2010), 693–712.
A. S. Merkurjev, A lower bound on the essential dimension of simple algebras, Algebra & Number Theory (to appear).
A. S. Merkurjev, I. Panin, A. R. Wadsworth, Index reduction formulas for twisted flag varieties, I, K-Theory 10 (1996), 517–596.
A. S. Merkurjev, I. Panin, A. R. Wadsworth, Index reduction formulas for twisted flag varieties, II, K-Theory 14 (1998), no. 2, 101–196.
А. С. Меркуръев, А. А. Суслин, К-когомологии многообразий Севери-Брауэра и гомомрфизм норменного вычета, Изв. АН СССР, сер. мат. 46 (1982), no. 5, 1011_1046. Engl. transl.: A. S. Merkurjev, A. A. Suslin, K -cohomology of Severi-Brauer varieties and the norm residue homomorphism, Math. USSR Izv. 21 (1983), no. 2, 307–340.
A. Meyer, Z. Reichstein, An upper bound on the essential dimension of a central simple algebra, J. Algebra (2009) (in press), doi:10.1016/j.jalgebra.2009.09.019.
J. S. Milne, The Tate Šafarevič group of a constant abelian variety, Invent. Math. 6 (1968), 91–105.
J. S. Milne, The Brauer group of a rational surface, Invent. Math. 11 (1970), 304–307.
J. S. Milne, On a conjecture of Artin and Tate, Ann. of Math. (2) 102 (1975), no. 3, 517–533.
J. S. Milne, Étale Cohomology, Princeton University Press, Princeton, NJ, 1980. Russian transl.: Дж. Милн, Зтальные когомологии, Мир, М., 1983.
V. K. Murty, D. Ramakrishnan, Period relations and the Tate conjecture for Hilbert modular surfaces, Invent. Math. 89 (1987), no. 2, 319–345.
T. Nakayama, Divisionalgebren über diskret bewerteten perfekten Körpen, J. Reine Angew. Math. 178 (1938), 11–13.
N. O. Nygaard, The Tate conjecture for ordinary K3 surfaces over finite fields, Invent. Math. 74 (1983), no. 2, 213–237.
N. Nygaard, A. Ogus, Tate’s conjecture for K3 surfaces of finite height, Ann. of Math. (2) 122 (1985), no. 3, 461–507.
R. Parimala, R. Sridharan, V. Suresh, A question on the discriminants of involutions of central division algebras, Math. Ann. 297 (1993), 575–580.
R. Parimala, R. Sridharan, V. Suresh, Hermitian analogue of a theorem of Springer, J. Algebra 243 (2001), no. 2, 780–789.
R. Parimala, V. Suresh, Isotropy of quadratic forms over function fields of p-adic curves, Publ. Math. Inst. Hautes Études Sci. 88 (1998), 129–150.
R. Parimala, V. Suresh, On the length of a quadratic form, in: Algebra and Number Theory, Hindustan Book Agency, Delhi, 2005, pp. 147–157.
R. Parimala, V. Suresh, The u-invariant of the function fields of p-adic curves, Ann. of Math. (2) 172 (2010), no. 2, 1391–1405.
R. S. Pierce, Associative Algebras, Graduate Texts in Mathematics, Vol. 88, Springer-Verlag, New York, 1982. Russian transl.: Р. Пирс, Ассоциативные алгебры, Мир, М., 1986.
В. П. Платонов, О проблеме Таннака-Артина, ДАН СССР 221 (1975), no. 5, 1038_1041. Engl. transl.: V. P. Platonov, On the Tannaka–Artin problem, Soviet Math. Dokl. 16 (1975), no. 2, 468–473.
B. Poonen, M. Stoll, The Cassels–Tate pairing on polarized abelian varieties, Ann. of Math. (2) 150 (1999), no. 3, 1109–1149.
C. Popescu, Torsion subgroups of Brauer groups and extensions of constant local degree for global function fields, J. Number Theory 115 (2005), 27–44.
C. Popescu, J. Sonn, A. Wadsworth, n-torsion of Brauer groups of abelian extensions, J. Number Theory 125 (2007), 26–38.
G. Prasad, A. S. Rapinchuk, Weakly commensurable arithmetic groups and isospectral locally symmetric spaces, Publ. Math. Inst. Hautes Études Sci. 109 (2009), no. 1, 113–184.
C. Procesi, Non-commutative affine rings, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I (8) 8 (1967), 237–255.
A. Rapinchuk, I. Rapinchuk, On division algebras having the same maximal subfields, Manuscripta Math. 132 (2010), nos. 3–4, 273–293.
Z. Reichstein, On a theorem of Hermite and Joubert, Canad. J. Math. 51 (1999), no. 1, 69–95.
Z. Reichstein, On the notion of essential dimension for algebraic groups, Transform. Groups 5 (2000), no. 3, 265–304.
Z. Reichstein, Essential dimension, in: Proceedings of the International Congress of Mathematicians 2010, World Scientific, Singapore, 2010.
Z. Reichstein, B. Youssin, Essential dimensions of algebraic groups and a resolution theorem for G-varieties, with an appendix by J. Kollár, E. Szabó, Canad. J. Math. 52 (2000), no. 5, 1018–1056.
Z. Reichstein, B. Youssin, Splitting fields of G-varieties, Pacific J. Math. 200 (2001), no. 1, 207–249.
L. J. Risman, Zero divisors in tensor products of division algebras, Proc. Amer. Math. Soc. 51 (1975), 35–36.
L. J. Risman, Cyclic algebras, complete fields, and crossed products, Israel J. Math. 28 (1977), nos. 1–2, 113–128.
P. Roquette, Isomorphisms of generic splitting fields of simple algebras, J. Reine Angew. Math. 214/215 (1964), 207–226.
P. Roquette, The Brauer–Hasse–Noether Theorem in Historical Perspective, Schr. Math.-Phys. Klasse Heidelb. Akad. Wiss., no. 15, Springer-Verlag, Berlin, 2004.
S. Rosset, Group extensions and division algebras, J. Algebra 53 (1978), no. 2, 297–303.
M. Rost, Computation of some essential dimensions, preprint, 2000.
M. Rost, J-P. Serre, J.-P. Tignol, La forme trace d’une algèbre simple centrale de degré 4, C. R. Math. Acad. Sci. Paris 342 (2006), 83–87.
L. H. Rowen, Central simple algebras, Israel J. Math. 29 (1978), nos. 2–3, 285–301.
L. H. Rowen, Division algebra counterexamples of degree 8, Israel J. Math. 38 (1981), 51–57.
L. H. Rowen, Cyclic division algebras, Israel J. Math. 41 (1982), no. 3, 213–234.
L. H. Rowen, Division algebras of exponent 2 and characteristic 2, J. Algebra 90 (1984), 71–83.
L. H. Rowen, Ring Theory, Vol. II, Pure Appl. Math., Vol. 128, Academic Press, Boston, MA, 1988.
L. H. Rowen, Ring Theory, Student edition, Academic Press, New York, 1991.
L. H. Rowen, Are p-algebras having cyclic quadratic extensions necessarily cyclic?, J. Algebra 215 (1999), 205–228.
L. H. Rowen, D. J. Saltman, Dihedral algebras are cyclic, Proc. Amer. Math. Soc. 84 (1982), no. 2, 162–164.
L. H. Rowen, D. J. Saltman, Prime to p extensions, Israel J. Math. 78 (1992), 197–207.
L. H. Rowen, D. J. Saltman, Semidirect product division algebras, Israel J. Math. 96 (1996), 527–552.
A. Ruozzi, Essential p-dimension of PGL n , J. Algebra 328 (2011), 488–494.
C. H. Sah, Symmetric bilinear forms and quadratic forms, J. Algebra 20 (1972), 144–160.
D. J. Saltman, Splittings of cyclic p-algebras, Proc. Amer. Math. Soc. 62 (1977), 223–228.
D. J. Saltman, Noncrossed product p-algebras and Galois p-extensions, J. Algebra 52 (1978), 302–314.
D. J. Saltman, Noncrossed products of small exponent, Proc. Amer. Math. Soc. 68 (1978), 165–168.
D. J. Saltman, Indecomposable division algebras, Comm. Algebra 7 (1979), no. 8, 791–817.
D. J. Saltman, On p-power central polynomials, Proc. Amer. Math. Soc. 78 (1980), 11–13.
D. J. Saltman, Review: John Dauns, “A Concrete Approach to Division Rings” and P.K. Draxl, “Skew Fields”, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 1, 214–221.
D. J. Saltman, The Brauer group and the center of generic matrices, J. Algebra 97 (1985), 53–67.
D. J. Saltman, Finite dimensional division algebras, in: Azumaya Algebras, Actions, and Modules, Contemp. Math., Vol. 124, Amer. Math. Soc., Providence, RI, 1992, pp. 203–214.
D. J. Saltman, Division algebras over p-adic curves, J. Ramanujan Math. Soc. 12 (1997), 25–47, see also the erratum [S98] and survey [B10].
D. J. Saltman, H 3 and generic matrices, J. Algebra 195 (1997), 387–422.
D. J. Saltman, Correction to division algebras over p-adic curves, J. Ramanujan Math. Soc. 13 (1998), 125–129.
D. J. Saltman, Lectures on Division Algebras, CBMS Reg. Conf. Ser. Math., Vol. 94, Amer. Math. Soc., Providence, RI, 1999.
D. J. Saltman, Amitsur and division algebras, in: Selected Papers of S. A. Amitsur with commentary, Part 2, Amer. Math. Soc., Providence, RI, 2001, pp. 109–115.
D. J. Saltman, Invariant fields of symplectic and orthogonal groups, J. Algebra 258 (2002), no. 2, 507–534.
D. J. Saltman, Cyclic algebras over p-adic curves, J. Algebra 314 (2007), 817–843.
D. J. Saltman, J.-P. Tignol, Generic algebras with involution of degree 8m, J. Algebra 258 (2002), no. 2, 535–542.
M. M. Schacher, Subfields of division rings, I, J. Algebra 9 (1968), 451–477.
M. M. Schacher, L. Small, Noncrossed products in characteristic p, J. Algebra 24 (1973), 100–103.
M. M. Schacher, J. Sonn, K -admissibility of A 6 and A 7, J. Algebra 145 (1992), no. 2, 333–338.
A. Schofield, Matrix invariants of composite size, J. Algebra 147 (1992), no. 2, 345–349.
A. Schofield, M. van den Bergh, The index of a Brauer class on a Brauer–Severi variety, Trans. Amer. Math. Soc. 333 (1992), no. 2, 729–739.
J-P. Serre, Cohomologie Galoisienne: Progrès et problèmes, Astérisque (1995), no. 227, 229–257, Séminaire Bourbaki, Vol. 1993/94, Exp. 783 (= Oe. 166).
J-P. Serre, Topics in Galois Theory, 2nd ed., with notes by Henri Darmon, Res. Notes Math., Vol. 1, A K Peters, Wellesley, MA, 2008.
A. S. Sivatski, Applications of Clifford algebras to involutions and quadratic forms, Comm. Algebra 33 (2005), no. 3, 937–951.
J. Sonn, ℚ-admissibility of solvable groups, J. Algebra 84 (1983), no. 2, 411–419.
V. Suresh, Bounding the symbol length in the Galois cohomology of function fields of p-adic curves, Comment. Math. Helv. 85 (2010), no. 2, 337–346.
A. A. Suslin, SK 1 of division algebras and Galois cohomology, Adv. Soviet Math. 4 (1991), 75–101.
J. Tate, Algebraic cycles and poles of zeta functions, in: Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), Harper & Row, New York, 1965, pp. 93–110. Russian transl.: Д. Тэйт, Алгебраические циклы иполюса дзета-функций, УМН 20 (1965), no. 6, 27–40.
J. Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134–144. Russian transl.: Д. Тэйт, Эндоморфизмы абелевых многообразий над конечными полями, Математика, сб. перев. 12 (1968), no. 6, 31–40.
J. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Séminaire Bourbaki, Vol. 1965/1966, Exp. 295–312, no. 306, W. A. Benjamin, New York, 1966, pp. 415–440. Russian transl.: Д. Тэйт, О гипотезах Бёрча и Свиннертона-Дайера и их геометрическом аналоге, Математика, сб. перев. 12 (1968), no. 6, 41–55.
O. Teichmüller, p-algebren, Deutsche Math. 1 (1936), 362–388.
O. Teichmüller, Zerfallende zyklische p-algebren, J. Reine Angew. Math. 176 (1937), 157–160.
J.-P. Tignol, On the length of decompositions of central simple algebras in tensor products of symbols, in: Methods in Ring Theory (Antwerp, 1983), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 129, Reidel, Dordrecht, 1984, pp. 505–516.
J.-P. Tignol, Cyclic and elementary abelian subfields of Malcev–Neumann division algebras, J. Pure Appl. Algebra 42 (1986), no. 2, 199–220.
J.-P. Tignol, On the corestriction of central simple algebras, Math. Z. 194 (1987), no. 2, 267–274.
J.-P. Tignol, La forme seconde trace d’une algèbre simple centrale de degré 4 de caractéristique 2, C. R. Math. Acad. Sci. Paris 342 (2006), 89–92.
J.-P. Tignol, S. A. Amitsur, Kummer subfields of Mal’cev–Neumann division algebras, Israel J. Math. 50 (1985), 114–144.
J.-P. Tignol, S. A. Amitsur, Symplectic modules, Israel J. Math. 54 (1986), no. 3, 266–290.
J.-P. Tignol, A. R. Wadsworth, Totally ramified valuations on finite-dimensional division algebras, Trans. Amer. Math. Soc. 302 (1987), no. 1, 223–250.
J. Tits, Classification of algebraic semisimple groups, in: Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math., Vol. IX, Amer. Math. Soc., Providence, RI, 1966, pp. 32–62. Russian transl.: Ж. Титс, Классификация полупростых алгебраических групп, Математика, сб. перев. 12 (1968), no. 2, 110–143.
J. Tits, Groupes de Whitehead de groupes algébriques simples sur un corps (d’après V. P. Platonov et al.), Lecture Notes in Math., Vol. 677, Exp. 505, Springer-Verlag, Berlin, 1978, pp. 218–236.
С. Л. Трегуб, О бирациональной зквивалеалентности многообрзий Брауэра-Севери, УМН 46 (1991), no. 6, 217–218. Engl. transl.: S. L. Tregub, Birational equivalence of Brauer-Severi manifolds, Russian Math. Surveys 46 (1991), no. 6, 229.
T. Urabe, The bilinear form of the Brauer group of a surface, Invent. Math. 125 (1996), no. 3, 557–585.
U. Vishne, Dihedral crossed products of exponent 2 are abelian, Arch. Math. (Basel) 80 (2003), no. 2, 119–122.
U. Vishne, Galois cohomology of fields without roots of unity, J. Algebra 279 (2004), 451–492.
V. E. Voskresenskiĭ, Algebraic Groups and their Birational Invariants, Transl. Math. Monogr., Vol. 179, Amer. Math. Soc., Providence, RI, 1998.
A. R. Wadsworth, Valuation theory on finite-dimensional division algebras, in: Valuation Theory and its Applications, Vol. I (Saskatoon, SK), Fields Inst. Commun., Vol. 32, Amer. Math. Soc., Providence, RI, 2002, pp. 385–449.
S. Wang, On the commutator group of a simple algebra, Amer. J. Math. 72 (1950), 323–334.
J. H. M. Wedderburn, On division algebras, Trans. Amer. Math. Soc. 22 (1921), 129–135.
A. Weil, Algebras with involution and the classical groups, J. Indian Math. Soc. 24 (1960), 589–623. Russian transl.: A. Вейль, Алгебры с инволюцией иклассичекие группы, Математика, сб. перев. 7 (1963), no. 4, 31–56.
E. Witt, Gegenbeispiel zum Normensatz, Math. Z. 39 (1934), 12–28.
E. Witt, Zyklische Körper und Algebren der characteristik p vom Grad p n: Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassen-körper der Charakteristik p, J. Reine Angew. Math. 176 (1937), 126–140.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Auel, A., Brussel, E., Garibaldi, S. et al. Open problems on central simple algebras. Transformation Groups 16, 219–264 (2011). https://doi.org/10.1007/s00031-011-9119-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00031-011-9119-8