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Central Simple Algebras and Galois Cohomology
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  • Philippe Gille, Institut Camille Jordan, Lyon, Tamás Szamuely, Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest
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The first comprehensive, modern introduction to the theory of central simple algebras over arbitrary fields, this book starts from the basics and reaches such advanced results as the Merkurjev–Suslin theorem, a culmination of work initiated by Brauer, Noether, Hasse and Albert, and the starting point of current research in motivic cohomology theory by Voevodsky, Suslin, Rost and others. Assuming only a solid background in algebra, the text covers the basic theory of central simple algebras, methods of Galois descent and Galois cohomology, Severi–Brauer varieties, and techniques in Milnor K-theory and K-cohomology, leading to a full proof of the Merkurjev–Suslin theorem and its application to the characterization of reduced norms. The final chapter rounds off the theory by presenting the results in positive characteristic, including the theorems of Bloch–Gabber–Kato and Izhboldin. This second edition has been carefully revised and updated, and contains important additional topics.

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[1] Albert, Adrian A. On the Wedderburn norm condition for cyclic algebras, Bull. Amer. Math. Soc. 37 (1931), 301–312.
[2] Albert, Adrian A. Normal division algebras of degree four over an algebraic field, Trans. Amer. Math. Soc. 34 (1932), 363–372.
[3] Albert, Adrian A. Simple algebras of degree pe over a centrum of characteristic p, Trans. Amer. Math. Soc. 40 (1936), 112–126.
[4] Albert, Adrian A. Structure of Algebras, American Mathematical Society Colloquium Publications, vol. XXIV, 1939.
[5] Albert, Adrian A. Tensor products of quaternion algebras, Proc. Amer. Math. Soc. 35 (1972), 65–66.
[1] Amitsur, Shimshon A. Generic splitting fields of central simple algebras, Ann. of Math. (2) 62 (1955), 8–43.
[2] Amitsur, Shimshon A. On central division algebras, Israel J. Math. 12 (1972), 408–420.
[1] Amitsur, Shimshon, Rowen, Louis H. and Tignol, Jean-Pierre Division algebras of degree 4 and 8 with involution, Israel J. Math. 33 (1979), 133–148.
[1] Amitsur, Shimshon and Saltman, David Generic Abelian crossed products and p-algebras, J. Algebra 51 (1978), 76–87.
[1] Antieau, Benjamin and Williams, Ben Unramified division algebras do not always contain Azumaya maximal orders, Invent. Math. 197 (2014), 47–56.
[2] Antieau, Benjamin and Williams, Ben The prime divisors of the period and index of a Brauer class, J. Pure Appl. Algebra 219 (2015), 2218–2224.
[3] Antieau, Benjamin and Williams, Ben Prime decomposition for the index of a Brauer class, Ann. Sc. Norm. Super. Pisa Cl. Sci., XVII(2017), 277–285.
[1] Arason, Jón Kristinn Cohomologische Invarianten quadratischer Formen, J. Algebra 36 (1975), 448–491.
[2] Arason, Jón Kristinn A proof of Merkurjev's theorem, in Quadratic and Hermitian forms (Hamilton, Ont., 1983), CMS Conf. Proc., 4, Amer. Math. Soc., Providence, 1984, 121–130.
[1] Artin, Emil Kennzeichnung des Körpers der reellen algebraischen Zahlen, Abh. Math. Sem. Hamburg 3 (1924), 319–323.
[1] Artin, Emil and Schreier, Otto Eine Kennzeichnung der reell abgeschlossenen Körper, Abh. Math. Sem. Hamburg 5 (1927), 225–231.
[1] Artin, Emil and Tate, John Class Field Theory, 2nd edition, Addison-Wesley, Redwood, 1990.
[1] Artin, Michael Brauer-Severi varieties, in Brauer groups in ring theory and algebraic geometry (Wilrijk, 1981), Lecture Notes in Math. 917, Springer-Verlag, Berlin-New York, 1982, 194–210.
[1] Artin, Michael and Mumford, David Some elementary examples of unirational varieties which are not rational, Proc. London Math. Soc. (3) 25 (1972), 75–95.
[1] Asok, Aravind Rationality problems and conjectures of Milnor and Bloch-Kato, Compositio Math. 149 (2013), 1312–1326.
[1] Atiyah, Michael Francis and Macdonald, Ian G. Introduction to Commutative Algebra, Addison-Wesley, Reading, 1969.
[1] Atiyah, Michael Francis and Wall, Charles Terence Clegg Cohomology of groups, in Algebraic Number Theory (J. W. S. Cassels and A. Fröhlich, eds.), Academic Press, London, 1967, 94–115.
[1] Auslander, Maurice and Brumer, Armand Brauer groups of discrete valuation rings, Indag. Math. 30 (1968), 286– 296.
[1] Ax, James A field of cohomological dimension 1 which is not C1, Bull. Amer. Math. Soc. 71 (1965), 717.
[2] Ax, James Proof of some conjectures on cohomological dimension, Proc. Amer. Math. Soc. 16 (1965), 1214–1221.
[1] Bass, Hyman, Milnor, John and Serre, Jean-Pierre Solution of the congruence subgroup problem for SLn (n ≥ 3) and Sp2n (n ≥ 2), Inst. Hautes Études Sci. Publ. Math. 33 (1967), 59–137.
[1] Bass, Hyman and Tate, John The Milnor ring of a global field, in Algebraic K-theory II, Lecture Notes in Math. 342, Springer-Verlag, Berlin, 1973, 349–446.
[1] Beauville, Arnaud Surfaces algébriques complexes, Astérisque No. 54, SociétéMathématique de France, Paris, 1978; English translation: Complex Algebraic Surfaces, London Mathematical Society Lecture Note Series, vol. 68, Cambridge University Press, 1983.
[1] Berhuy, Grégory and Frings, Christoph On the second trace form of central simple algebras in characteristic two, Manuscripta Math. 106 (2001), 1–12.
[1] Bloch, Spencer K2 and algebraic cycles, Ann. of Math. (2) 99 (1974), 349–379.
[2] Bloch, Spencer Lectures on Algebraic Cycles, Duke University Mathematics Series IV, Duke University, Durham, 1980. Second edition: New Mathematical Monographs, vol. 16, Cambridge University Press, 2010.
[3] Bloch, Spencer Torsion algebraic cycles, K2, and Brauer groups of function fields, in The Brauer Group (Les Plans-sur-Bex, 1980), Lecture Notes in Math. 844, Springer-Verlag, Berlin, 1981, 75–102.
[1] Bloch, Spencer and Kato, Kazuya p-adic étale cohomology, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 107–152.
[1] Bogomolov, Fedor A. The Brauer group of quotient spaces of linear representations (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), 485–516, 688; English translation in Math. USSR-Izv. 30 (1988), 455–485.
[2] Bogomolov, Fedor A. Brauer groups of the fields of invariants of algebraic groups (Russian), Mat. Sb. 180 (1989), 279–293; English translation in Math. USSR-Sb. 66 (1990), 285–299.
[1] Brauer, Richard Untersuchungen über die arithmetischen Eigenschaften von Gruppen linearer Substitutionen I, Math. Zeit. 28 (1928), 677–696; II, ibid. 31 (1930), 733–747.
[2] Brauer, Richard Über die algebraische Struktur von Schiefkörpern, J. reine angew. Math. 166 (1932), 241–252.
[1] Brauer, Richard, Hasse, Helmut and Noether, Emmy, Beweis eines Hauptsatzes in der Theorie der Algebren, J. reine angew. Math. 167 (1932), 399–404.
[1] Brussel, Eric Noncrossed products and nonabelian crossed products over Q(t) and ((t)), Amer. J. Math. 117 (1995), 377–393.
[1] Cartan, Henri and Eilenberg, Samuel Homological Algebra, Princeton University Press, Princeton, 1956.
[1] Cartier, Pierre Questions de rationalité des diviseurs en géométrie algébrique, Bull. Soc. Math. France 86 (1958), 177–251.
[1] Cassels, John William Scott and Fröhlich, Albrecht (eds.) Algebraic Number Theory, Academic Press, London, 1967.
[1] Clemens, Herbert and Griffiths, Phillip The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281–356.
[1] Châtelet, François Variations sur un thème de H. Poincaré, Ann. Sci. Éc. Norm. Sup. III. Sér. 61 (1944), 249–300.
[1] Chevalley, Claude Démonstration d'une hypothèse de M. Artin, Abh. Math. Semin. Hamb. Univ. 11 (1935), 73–75.
[1] Chu, Huah and Kang, Ming-chang Rationality of p-group actions, J. Algebra 237 (2001), 673–690.
[1] Chu, Huah, Hu, Shou-Yen, Kang, Ming-chang and Prokhorov, Yuri Noether's problem for groups of order 32, J. Algebra 320 (2008), 3022– 3035.
[1] Colliot-Thélène, Jean-Louis Hilbert's Theorem 90 for K2, with application to the Chow groups of rational surfaces, Invent. Math. 71 (1983), 1–20.
[2] Colliot-Thélène, Jean-Louis Les grands thèmes de François Châtelet, Enseign. Math. (2) 34 (1988), 387–405.
[3] Colliot-Thélène, Jean-Louis Cycles algébriques de torsion et K-théorie algébrique, in Arithmetic algebraic geometry (Trento, 1991), Lecture Notes in Math. 1553, Springer- Verlag, Berlin, 1993, 1–49.
[4] Colliot-Thélène, Jean-Louis Cohomologie galoisienne des corps valués discrets henséliens, d'après K. Kato et S. Bloch, in Algebraic K-Theory and Its Applications (Trieste, 1997) (H., Bass, A., Kuku, C., Pedrini, eds.), World Scientific, River Edge, 1999, 120–163.
[5] Colliot-Thélène, Jean-Louis Fields of cohomological dimension one versus C1-fields, in Algebra and Number Theory (R., Tandon, ed.), Hindustan Book Agency, New Delhi, 2005, 1–6.
[1] Colliot-Thélène, Jean-Louis, Hoobler, Raymond and Kahn, Bruno The Bloch-Ogus–Gabber theorem, in Algebraic K-theory (Toronto, 1996), Fields Inst. Commun., vol. 16, Amer. Math. Soc., Providence, 1997, 31–94.
[1] Colliot-Thélène, Jean-Louis, and Madore, David A. Surfaces de del Pezzo sans point rationnel sur un corps de dimension cohomologique un, J. Inst. Math. Jussieu 3 (2004), 1–16.
[1] Colliot-Thélène, Jean-Louis and Ojanguren, ManuelVariétés unirationnelles non rationnelles: au-delà de l'exemple d'Artin et Mumford, Invent. Math. 97 (1989), 141–158.
[1] Colliot-Thélène, Jean-Louis, Ojanguren, Manuel and Parimala, Raman Quadratic forms over fraction fields of two-dimensional Henselian rings and Brauer groups of related schemes, in Algebra, Arithmetic and Geometry, Tata Inst. Fund. Res. Stud. Math., vol. 16, Bombay, 2002, 185–217.
[1] Colliot-Thélène, Jean-Louis and Sansuc, Jean-Jacques The rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group), in Proceedings of the International Colloquium on Algebraic Groups and Homogeneous Spaces (Mumbai, 2004) (V., Mehta, ed.), TIFR Mumbai, Narosa Publishing House, 2007, 113–186.
[1] Colliot-Thélène, Jean-Louis, Sansuc, Jean-Jacques and Soulé, Christophe Torsion dans le groupe de Chow de codimension deux, Duke Math. J. 50 (1983), 763–801.
[1] Demazure, Michel and Gabriel, Pierre Groupes algébriques I, Masson, Paris and North-Holland, Amsterdam, 1970.
[1] Dennis, R. Keith and Stein, Michael R. K2 of discrete valuation rings, Adv. in Math. 18 (1975), 182–238.
[1] Dickson, Lawrence J. Linear algebras, Trans. Amer. Math. Soc. 13 (1912), 59–73.
[1] Dieudonné, Jean Les déterminants sur un corps non commutatif, Bull. Soc. Math. France 71 (1943), 27–45.
[2] Dieudonné, Jean Sur une généralisation du groupe orthogonal à quatre variables, Arch.Math. 1 (1949), 282–287.
[1] Draxl, Peter Skew Fields, London Mathematical Society Lecture Note Series, vol. 81, Cambridge University Press, 1983.
[1] Elman, Richard, Karpenko, Nikita and Merkurjev, Alexander The Algebraic and Geometric Theory of Quadratic Forms, Colloquium Publications, vol. 56, American Mathematical Society, Providence, 2008.
[1] Elman, Richard and Lam, Tsit Yuen Pfister forms and K-theory of fields, J. Algebra 23 (1972), 181–213.
[1] Faddeev, Dmitri K. Simple algebras over a field of algebraic functions of one variable (Russian), Trudy Mat. Inst. Steklova 38 (1951), 321–344; English translation in Amer. Math. Soc. Transl. (2) 3 (1956), 15–38.
[2] Faddeev, Dmitri K. On the theory of homology in groups (Russian), Izv. Akad. Nauk SSSR. Ser. Mat. 16 (1952), 17–22.
[3] Faddeev, Dmitri K. On the theory of algebras over the field of algebraic functions of one variable (Russian), Vestnik Leningrad. Univ. Ser. Mat. Meh. Astr. 12 (1957), 45–51.
[1] Fesenko, Ivan B. and Vostokov, Sergey V. Local Fields and Their Extensions, 2nd edition, Translations of Mathematical Monographs, vol. 121, American Mathematical Society, Providence, 2002.
[1] Fischer, Ernst Die Isomorphie der Invariantenkörper der endlicher Abelschen Gruppen linearer Transformationen, Nachr. Akad. Wiss. Göttingen Math.-Phys. 1915, 77–80.
[2] Fischer, Ernst Zur Theorie der endlichen Abelschen Gruppen, Math. Ann. 77 (1915), 81–88.
[1] Florence, Mathieu On the symbol length of p-algebras, Compositio Math. 149 (2013), 1353– 1363.
[1] Fulton, William Intersection Theory, 2nd edition, Springer-Verlag, Berlin, 1998.
[1] Garibaldi, Skip, Merkurjev, Alexander and Serre, Jean-Pierre Cohomological invariants in Galois cohomology, University Lecture Series 28, American Mathematical Society, Providence, RI, 2003.
[1] Geisser, Thomas and Levine, Marc The K-theory of fields in characteristic p, Invent. Math. 139 (2000), 459–493.
[1] Gerstenhaber, Murray On infinite inseparable extensions of exponent one, Bull. Amer. Math. Soc. 71 (1965), 878–881.
[1] Graber, Tom, Harris, Joe and Starr, Jason Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003), 57–67.
[1] Greenberg, Marvin J. Rational points in Henselian discrete valuation rings, Inst. Hautes Études Sci. Publ. Math. 31 (1966), 59–64.
[2] Greenberg, Marvin J. Lectures On Forms In Many Variables, W. A. Benjamin, New York-Amsterdam, 1969.
[1] Grothendieck, Alexander Technique de descente et théorèmes d'existence en géométrie algébrique I: Généralités. Descente par morphismes fidèlement plats, Sém. Bourbaki, exp.190 (1960); reprinted by Société Mathématique de France, Paris, 1995.
[2] Grothendieck, Alexander Revêtements étales et groupe fondamental (SGA 1), Lecture Notes in Mathematics, vol. 224, Springer-Verlag, Berlin, 1971; reprinted as vol. 3 of Documents Mathématiques, Société Mathématique de France, Paris, 2003.
[3] Grothendieck, Alexander Le groupe de Brauer I, II, III, in Dix exposés sur la cohomologie des schémas, North-Holland, Amsterdam/Masson, Paris, 1968, 46–188.
[4] Grothendieck, Alexander Éléments de géométrie algébrique IV: Étude locale des schémas et des morphismes de schémas, 2e partie, Inst. Hautes Études Sci. Publ. Math. 24, (1965).
[1] Hartshorne, Robin Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, Springer- Verlag, New York-Heidelberg, 1977.
[1] Heuser, Ansgar Über den Funktionenkörper der Normfläche einer zentral einfachen Algebra, J. reine angew. Math. 301 (1978), 105–113.
[1] Hochschild, Gerhard Cohomology of restricted Lie algebras, Amer. J. Math. 76 (1954), 555–580.
[2] Hochschild, Gerhard Simple algebras with purely inseparable splitting fields of exponent 1, Trans. Amer. Math. Soc. 79 (1955), 477–489.
[1] Hochschild, Gerhard and Nakayama, Tadasi Cohomology in class field theory, Ann. of Math. 55 (1952), 348–366.
[1] Hoshi, Akinari, Kang, Ming-chang and KunyavskiĬ, Boris Noether's problem and unramified Brauer groups, Asian J. Math. 17 (2013), 689–713.
[1] Iskovskih, Vasiliy A. and Manin, Yuri I. Three-dimensional quartics and counterexamples to the Lüroth problem (Russian), Mat. Sb. 86 (128) (1971), 140–166.
[1] Illusie, Luc Complexe de de Rham-Witt et cohomologie cristalline, Ann. Sci. École Norm. Sup. (4) 12 (1979), 501–661.
[2] Illusie, Luc Frobenius et dégénérescence de Hodge, in J.-P., Demailly et al., Introduction à la théorie de Hodge, Soc. Math. France, Paris, 1996, 113–168.
[1] Izhboldin, Oleg T. On p-torsion in for fields of characteristic p, Adv. Soviet Math. 4 (1991), 129–144.
[1] Jacob, Bill Indecomposable division algebras of prime exponent, J. reine angew.Math. 413 (1991), 181–197.
[1] Jacobson, Nathan Abstract derivations and Lie algebras, Trans. Amer. Math. Soc. 42 (1937), 206–224.
[2] Jacobson, Nathan Basic Algebra I-II, 2nd edition, W. H. Freeman & Co., New York, 1989.
[3] Jacobson, Nathan Finite-dimensional Division Algebras over Fields, Springer-Verlag, Berlin, 1996.
[1] Jahnel, Jörg The Brauer-Severi variety associated with a central simple algebra: a survey, available from the server, preprint No. 52, 2000.
[1] de Jong, Aise Johan The period-index problem for the Brauer group of an algebraic surface, Duke Math. J. 123 (2004), 71–94.
[1] de Jong, Aise Johan and Starr, Jason Every rationally connected variety over the function field of a curve has a rational point, Amer. J. Math. 125 (2003), 567–580.
[1] Kahn, Bruno La conjecture de Milnor (d'après V. Voevodsky), Séminaire Bourbaki, exp. 834, Astérisque 245 (1997), 379–418.
[2] Kahn, Bruno Motivic cohomology of smooth geometrically cellular varieties, in Algebraic K-theory (W., Raskind and C., Weibel, eds.), Proc. Sympos. Pure Math. 67, Amer. Math. Soc., Providence, 1999, 149–174.
[3] Kahn, Bruno Quelques remarques sur le u-invariant, Sém. Théorie des Nombres de Bordeaux 2 (1990), 155–161.
[1] Kang, Ming-Chang Constructions of Brauer-Severi varieties and norm hypersurfaces, Canad. J. Math. 42 (1990), 230–238.
[1] Kato, Kazuya A generalization of local class field theory by using K-groups I, J. Fac. Sci. Univ. Tokyo 26 (1979), 303–376; II, J. Fac. Sci. Univ. Tokyo 27 (1980), 603–683.
[2] Kato, Kazuya Galois cohomology of complete discrete valuation fields, in Algebraic K-theory II (Oberwolfach, 1980), Lecture Notes in Math., 967, Springer- Verlag, Berlin-New York, 1982, 215–238.
[3] Kato, Kazuya Residue homomorphisms in Milnor K-theory, in Galois groups and their representations (Nagoya, 1981), Adv. Stud. Pure Math., 2, North-Holland, Amsterdam, 1983, 153–172.
[4] Kato, Kazuya Milnor K-theory and the Chow group of zero cycles, in Applications of Algebraic K-Theory to Algebraic Geometry and Number Theory (S., Bloch et al., eds.), Contemp. Math., 55/2, Amer. Math. Soc., Providence, 1986, 241–253.
[1] Kato, Kazuya and Kuzumaki, Takako The dimension of fields and algebraic K-theory, J. Number Theory 24 (1986), 229–244.
[1] Katz, Nicholas M. Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Inst. Hautes Études Sci. Publ.Math. 39 (1970), 175–232.
[2] Katz, Nicholas M. Algebraic solutions of differential equations (p-curvature and the Hodge filtration), Invent. Math. 18 (1972), 1–118.
[1] Kersten, Ina Brauergruppen von Körpern, Vieweg, Braunschweig, 1990.
[2] Kersten, Ina Noether's problem and normalization, Jahresber. Deutsch. Math.-Verein. 100 (1998), 3–22.
[1] Kerz, Moritz The Gersten conjecture for Milnor K-theory, Invent. Math. 175 (2009), 1–33.
[1] Klingen, Norbert A short remark on the Merkurjev-Suslin theorem, Arch. Math. 48 (1987), 126–129.
[1] Kneser, Martin Konstruktive Lösung p-adischer Gleichungssysteme, Nachr. Akad. Wiss. Göttingen Math.-Phys. (1978), 67–69.
[1] Knus, Max-Albert Sur la forme d'Albert et le produit tensoriel de deux algèbres de quaternions, Bull. Soc. Math. Belg. Sér. B 45 (1993), 333–337.
[1] Knus, Max-Albert, Merkurjev, Alexander, Rost, Markus and Tignol, Jean- Pierre The Book of Involutions, American Mathematical Society Colloquium Publications, vol. 44, American Mathematical Society, Providence, 1998.
[1] Kollár, János Severi-Brauer varieties; a geometric treatment, preprint arXiv:1606. 04368, 2016.
[1] Krull, Wolfgang Galoissche Theorie der unendlichen algebraischen Erweiterungen, Math. Ann. 100 (1928), 687–698.
[1] Lam, Tsit-Yuen Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics, vol. 67, American Mathematical Society, Providence, 2005.
[1] Lang, Serge On quasi-algebraic closure, Ann. of Math. (2) 55 (1952), 373–390.
[2] Lang, Serge Algebraic groups over finite fields, Amer. J. Math. 78 (1956), 555–563.
[3] Lang, Serge Algebra, 3rd edition, Addison-Wesley, Redwood, 1993.
[1] Lenstra, Hendrik W. Rational functions invariant under a finite abelian group, Invent. Math. 25 (1974), 299–325.
[2] Lenstra, Hendrik W. K2 of a global field consists of symbols, in Algebraic K-theory, Lecture Notes in Math., vol. 551, Springer-Verlag, Berlin, 1976, pp. 69–73.
[1] Lichtenbaum, Stephen The period-index problem for elliptic curves, Amer. J. Math. 90 (1968), 1209–1223.
[1] Lieblich, Max Twisted sheaves and the period-index problem, Compositio Math. 144 (2008), 1–31.
[2] Lieblich, Max Period and index in the Brauer group of an arithmetic surface (with an appendix by Daniel Krashen), J. reine angew. Math. 659 (2011), 1–41.
[3] Lieblich, Max The period-index problem for fields of transcendence degree 2, Ann. of Math. 182 (2015), 391–427.
[1] Mammone, Pascal and Merkurjev, Alexander S. On the corestriction of pn-symbol, Israel J. Math. 76 (1991), 73–79.
[1] Mammone, Pascal and Tignol, Jean-Pierre Dihedral algebras are cyclic, Proc. Amer. Math. Soc. 101 (1987), 217–218.
[1] Matsumura, Hideyuki Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, 1989.
[1] Matzri, Eliyahu All dihedral division algebras of degree five are cyclic, Proc. Amer. Math. Soc. 136 (2008), 1925–1931.
[1] McKinnie, Kelly Noncyclic and indecomposable p-algebras, PhD thesis, University of Texas at Austin, 2006.
[1] Merkurjev, Alexander S. On the norm residue symbol of degree 2 (Russian), Dokl. Akad. Nauk SSSR 261 (1981), 542–547.
[2] Merkurjev, Alexander S. K2 of fields and the Brauer group, in Applications of Algebraic KTheory to Algebraic Geometry and Number Theory (S., Bloch et al., eds.), Contemp. Math., vol. 55/1, Amer. Math. Soc., Providence, 1986, 529–546.
[3] Merkurjev, Alexander S. Kaplansky's conjecture in the theory of quadratic forms (Russian), Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 175 (1989), 75–89, 163–164; English translation in J. Soviet Math. 57 (1991), 3489–3497.
[4] Merkurjev, Alexander S. K-theory of simple algebras, in K-theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras (B., Jacob and J., Rosenberg, eds.), Proc. Sympos. Pure Math., vol. 58, Part 1, Amer. Math. Soc., Providence, 1995, 65–83.
[5] Merkurjev, Alexander S. On the norm residue homomorphism of degree two, Proceedings of the St. Petersburg Mathematical Society, vol. XII, Amer. Math. Soc. Transl. Ser. 2, vol. 219, American Mathematical Society, Providence, 2006, 103–124.
[6] Merkurjev, Alexander S. Brauer groups of fields, Comm. Algebra 11 (1983), 2611–2624.
[1] Merkurjev, Alexander S. and Suslin, Andrei A. K-cohomology of Severi-Brauer varieties and the norm residue homomorphism (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 1011–1046, 1135–1136.
[2] Merkurjev, Alexander S. and Suslin, Andrei A. Norm residue homomorphism of degree three (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), 339–356; English translation in Math. USSR Izv. 36 (1991), 349–367.
[3] Merkurjev, Alexander S. and Suslin, Andrei A. The group K3 for a field (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), 522–545; English translation in Math. USSR Izv. 36 (1991), 541– 565.
[1] Milne, James S. Duality in the flat cohomology of a surface, Ann. Sci. École Norm. Sup. 9 (1976), 171–201.
[2] Milne, James S. Étale Cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, 1980.
[3] Milne, James S. Arithmetic Duality Theorems, Academic Press, Boston, 1986.
[4] Milne, James S. Jacobian varieties, in Arithmetic Geometry (G., Cornell and J. H., Silverman, eds.), Springer-Verlag, New York, 1986, 167–212.
[1] Milnor, John W. Algebraic K-theory and quadratic forms, Invent. Math. 9 (1969/1970), 318–344.
[2] Milnor, John W. Introduction to algebraic K-theory, Annals of Mathematics Studies No. 72, Princeton University Press, 1971.
[1] Mumford, David The Red Book of Varieties and Schemes, Lecture Notes in Mathematics 1358, Springer-Verlag, Berlin, 1988.
[1] Murre, Jacob P. Applications of algebraic K-theory to the theory of algebraic cycles, in Algebraic geometry, Sitges (Barcelona), 1983, Lecture Notes in Math. 1124, Springer-Verlag, Berlin, 1985, 216–261.
[1] Neukirch, Jürgen Algebraic Number Theory, Grundlehren der Mathematischen Wissenschaften, vol. 322, Springer-Verlag, Berlin, 1999.
[1] Neukirch, Jürgen, Schmidt, Alexander and Wingberg, Kay Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, vol. 323, Springer-Verlag, Berlin, 2000.
[1] Peyre, Emmanuel Unramified cohomology of degree 3 and Noether's problem, Invent. Math. 171 (2008), 191–225.
[1] Pfister, Albrecht Quadratic Forms with Applications to Algebraic Geometry and Topology, London Mathematical Society Lecture Note Series, vol. 217, Cambridge University Press, 1995.
[1] Pierce, Richard Associative Algebras, Graduate Texts in Mathematics, vol. 88, Springer- Verlag, New York-Berlin, 1982.
[1] Platonov, Vladimir P. On the Tannaka–Artin problem, Dokl. Akad. Nauk SSSR 221 (1975), 1038– 1041.
[1] Quillen, Daniel Higher algebraic K-theory I, in Algebraic K-theory I: Higher K-theories, Lecture Notes in Math. 341, Springer-Verlag, Berlin 1973, 85–147.
[1] Rehmann, Ulf, Tikhonov, Sergey V. and YanchevskiĬ, Vyacheslav I. Symbol algebras and the cyclicity of algebras after a scalar extension (Russian), Fundam. Prikl. Mat. 14 (2008), 193–209; English translation in J. Math. Sci. (N. Y.) 164 (2010), 131–142.
[1] Ribes, Luis and Zalesskii, Pavel Profinite groups, Second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 40, Springer-Verlag, Heidelberg, 2010.
[1] Rieffel, Marc A. A generalWedderburn theorem, Proc. Nat. Acad. Sci. USA 54 (1965), 1513.
[1] Roquette, Peter On the Galois cohomology of the projective linear group and its applications to the construction of generic splitting fields of algebras, Math. Ann. 150 (1963), 411–439.
[2] Roquette, Peter Splitting of algebras by function fields of one variable, Nagoya Math. J. 27 (1966), 625–642.
[3] Roquette, Peter Class Field Theory in characteristic p: Its origin and development, in Class Field Theory - Its Centenary and Prospect (K., Miyake, ed.), Advanced Studies in Pure Mathematics 30, Tokyo, 2000, 549–631.
[4] Roquette, Peter The Brauer-Hasse-Noether Theorem in Historical Perspective, Schriften der Mathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften, vol. 15, Springer-Verlag, Berlin, 2005.
[1] Rosset, Shmuel and Tate, John A reciprocity law for K2-traces, Comment. Math. Helv. 58 (1983), 38–47.
[1] Rost, Markus Chow groups with coefficients, Doc. Math. 1 (1996), 319–393.
[2] Rost, Markus The chain lemma for Kummer elements of degree 3, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), 185–190.
[3] Rost, Markus Hilbert's Satz 90 for K3 for quadratic extensions, preprint, 1988, available from the author's homepage.
[4] Rost, Markus Chain lemma for splitting fields of symbols, preprint, 1998, available from the author's homepage.
[1] Rowen, Louis Halle Cyclic division algebras, Israel J. Math. 41 (1982), 213–234; Correction, Israel J. Math. 43 (1982), 277–280.
[2] Rowen, Louis Halle Ring Theory II, Academic Press, Boston, 1988.
[3] Rowen, Louis Halle Are p-algebras having cyclic quadratic extensions necessarily cyclic? J. Algebra 215 (1999), 205–228.
[1] Rowen, Louis Halle and Saltman, David J. Dihedral algebras are cyclic, Proc. Amer. Math. Soc. 84 (1982), 162–164.
[1] Saltman, David J. Generic Galois extensions and problems in field theory, Adv. in Math. 43 (1982), 250–283.
[2] Saltman, David J. Noether's problem over an algebraically closed field, Invent. Math. 77 (1984), 71–84.
[3] Saltman, David J. Lectures on Division Algebras, American Mathematical Society, Providence, 1999.
[4] Saltman, David J. Division algebras over p-adic curves, J. Ramanujan Math. Soc. 12 (1997), 25–47; correction, ibid. 13 (1998), 125–129.
[1] Scharlau, Winfried Über die Brauer-Gruppe eines algebraischen Funktionenkörpers in einer Variablen, J. reine angew. Math. 239/240 (1969), 1–6.
[2] Scharlau, Winfried Quadratic and Hermitian Forms, Grundlehren der Mathematischen Wissenschaften, vol. 270, Springer-Verlag, Berlin, 1985.
[1] Segre, Beniamino Questions arithmétiques sur les variétés algébriques, Colloques Internat. CNRS, vol. 24 (1950), 83–91.
[1] Serre, Jean-Pierre Sur la topologie des variétés algébriques en caractéristique p, in Symposium internacional de topologia algebraica, Mexico City, 1958, 24–53.
[2] Serre, Jean-Pierre Corps locaux, Hermann, Paris, 1962; English translation: Local Fields, Springer-Verlag, 1979.
[3] Serre, Jean-Pierre Lectures on the Mordell-Weil Theorem, Vieweg, Braunschweig, 1989.
[4] Serre, Jean-Pierre Cohomologie Galoisienne, 5e éd., révisée et complétée. Lecture Notes in Mathematics 5, Springer-Verlag, Berlin, 1994; English translation: Galois Cohomology, Springer-Verlag, Berlin, 2002.
[1] Seshadri, Conjeerveram Srirangachari L'opérateur de Cartier. Applications, Séminaire Chevalley, année 1958/59, exposé 6.
[1] Severi, Francesco Un nuovo campo di ricerche nella geometria sopra una superficie e sopra una varietà algebrica, Mem. Accad. Ital., Mat. 3 (1932), 1–52.
[1] Shafarevich, Igor R. On the Lüroth problem (Russian), Trudy Mat. Inst. Steklov 183 (1990), 199–204; English translation in Proc. Steklov Inst. Math. (1991), No. 4, 241–246.
[2] Shafarevich, Igor R. Basic Algebraic Geometry I-II, Springer-Verlag, Berlin, 1994.
[1] Shatz, Stephen S. Profinite Groups, Arithmetic, and Geometry, Annals of Mathematics Studies, No. 67, Princeton University Press, 1972.
[1] Soulé, Christophe K2 et le groupe de Brauer (d'après A. S., Merkurjev et A. A., Suslin), Séminaire Bourbaki, exp. 601, Astérisque 105–106 (1983), Soc. Math. France, Paris, 79–93.
[1] Speiser, Andreas Zahlentheoretische Sätze aus der Gruppentheorie, Math. Zeit. 5 (1919), 1–6.
[1] Springer, Tonny A. Linear Algebraic Groups, 2nd edition, Progress in Mathematics, vol. 9, Birkhäuser, Boston, MA, 1998.
[1] Sridharan, Ramaiyengar (in collaboration with Raman Parimala) 2-Torsion in Brauer Groups: A Theorem of Merkurjev, notes from a course held at ETH Zürich in 1984/85, available from the homepage of M.-A. Knus.
[1] Srinivas, Vasudevan Algebraic K-Theory, 2nd edition, Progress in Mathematics, vol. 90, Birkhäuser, Boston, 1996.
[1] Suslin, Andrei A. Algebraic K-theory and the norm residue homomorphism (Russian), in Current problems in mathematics 25 (1984), 115–207.
[2] Suslin, Andrei A. Torsion in K2 of fields, K-Theory 1 (1987), 5–29.
[3] Suslin, Andrei A. SK1 of division algebras and Galois cohomology, Adv. Soviet Math. 4, Amer. Math. Soc., Providence, 1991, 75–99.
[1] Suslin, Andrei A. and Joukhovitski, Seva Norm varieties, J. Pure Appl. Algebra 206 (2006), 245–276.
[1] Suslin, Andrei A. and Voevodsky, Vladimir Bloch-Kato conjecture and motivic cohomology with finite coefficients, in The Arithmetic and Geometry of Algebraic Cycles (B., Brent Gordon et al., eds.), Kluwer, Dordrecht, 2000, 117–189.
[1] Suzuki, Michio Group Theory I, Grundlehren der Mathematischen Wissenschaften, vol. 247, Springer-Verlag, New York, 1982.
[1] Swan, Richard G. Invariant rational functions and a problem of Steenrod, Invent. Math. 7 (1969), 148–158.
[2] Swan, Richard G. Noether's problem in Galois theory, in Emmy Noether in Bryn Mawr, Springer-Verlag, New York-Berlin, 1983, 21–40.
[3] Swan, Richard G. Higher algebraic K-theory, in K-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras (B., Jacob and J., Rosenberg, eds.), Proc. Sympos. Pure Math., vol. 58, Part I, Amer. Math. Soc., Providence, 1995, 247–293.
[1] Tabuada, Gonçalo and van den Bergh, Michel Noncommutative motives of Azumaya algebras, J. Inst. Math. Jussieu 14 (2015), 379–403.
[1] Tate, John Genus change in inseparable extensions of function fields, Proc. Amer. Math. Soc. 3 (1952), 400–406.
[2] Tate, John Global class field theory, in Algebraic Number Theory (J.W. S., Cassels and A., Fröhlich, eds.), Academic Press, London, 1967, 162–203.
[3] Tate, John Symbols in arithmetic, in Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, Gauthier-Villars, Paris, 1971, 201–211.
[4] Tate, John Relations between K2 and Galois cohomology, Invent. Math. 36 (1976), 257–274.
[5] Tate, John On the torsion in K2 of fields, in Algebraic number theory (Kyoto, 1976), Japan Soc. Promotion Sci., Tokyo, 1977, 243–261.
[1] Teichmüller, Oswald p-Algebren, Deutsche Math. 1 (1936), 362–388.
[1] Tignol, Jean-Pierre Algèbres indécomposables d'exposant premier, Adv. in Math. 65 (1987), 205–228.
[2] Tignol, Jean-Pierre On the corestriction of central simple algebras, Math. Z. 194 (1987), 267–274.
[1] Tignol, Jean-Pierre and Wadsworth, Adrian Value Functions on Simple Algebras, and Associated Graded Rings, Springer-Verlag, Heidelberg, New York, 2015.
[1] Tits, Jacques Sur les produits tensoriels de deux algèbres de quaternions, Bull. Soc. Math. Belg. Sér. B 45 (1993), 329–331.
[1] Tregub, Semion L. Birational equivalence of Brauer-Severi manifolds, Uspekhi Mat. Nauk 46 (1991), 217–218; English translation in Russian Math. Surveys 46 (1992), 229.
[1] Tsen, Chiung-Tse Divisionsalgebren über Funktionenkörpern, Nachr. Akad. Wiss. Göttingen Math.-Phys. 1933, 335–339.
[1] Voevodsky, Vladimir Motivic cohomology with Z/2-coefficients, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 59–104.
[2] Voevodsky, Vladimir On motivic cohomology with Z /l-coefficients, Ann. of Math. 174 (2011), 401–438.
[1] Voevodsky, Vladimir, Suslin, Andrei A. and Friedlander, Eric M. Cycles, Transfers, and Motivic Homology Theories, Ann. of Math. Studies, vol. 143, Princeton Univ. Press, 2000.
[1] VoskresenskiĬ, Valentin E. On the question of the structure of the subfield of invariants of a cyclic group of automorphisms of the field Q(x1, · · ·, xn) (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 366–375; English translation in Math. USSR Izv.4 (1971), 371–380.
[2] VoskresenskiĬ, Valentin E. Algebraic Groups and Their Birational Invariants, Translations of Mathematical Monographs, vol. 179, American Mathematical Society, Providence, 1998.
[1] Wadsworth, Adrian Merkurjev's elementary proof of Merkurjev's theorem, in Applications of Algebraic K-Theory to Algebraic Geometry and Number Theory (S., Bloch et al., eds.), Contemp. Math., vol. 55/2, Amer. Math. Soc., Providence, 1986, 741–776.
[1] van der Waerden, Bartel Leendert Algebra I (7te Auflage) II (5te Auflage), Springer-Verlag, Berlin, 1966–67. English translation: Springer-Verlag, New York, 1991.
[1] Wang, Shianghaw On the commutator group of a simple algebra, Amer. J. Math. 72 (1950), 323–334.
[1] Warning, Ewald Bemerkung zur vorstehenden Arbeit von Herrn Chevalley, Abh. Math. Semin. Hamb. Univ. 11 (1935), 76–83.
[1] Wedderburn, Joseph Henry Maclagan A theorem on finite algebras, Trans. Amer. Math. Soc. 6 (1905), 349–352.
[2] Wedderburn, Joseph Henry Maclagan On hypercomplex numbers, Proc. London Math. Soc. 6 (1908), 77–118.
[3] Wedderburn, Joseph Henry Maclagan On division algebras, Trans. Amer. Math. Soc. 22 (1921), 129–135.
[1] Weibel, Charles An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, 1994.
[2] Weibel, Charles The K-book: Introduction to Algebraic K-Theory, Graduate Studies in Mathematics, vol. 145, American Mathematical Society, Providence, 2013.
[1] Weil, André Sur la théorie du corps de classes, J. Math. Soc. Japan 3 (1951), 1–35.
[2] Weil, André The field of definition of a variety, Amer. J. Math. 78 (1956), 509–524.
[3] Weil, André Basic Number Theory, Basic number theory, 3rd edition, Grundlehren der Mathematischen Wissenschaften, vol. 144, Springer-Verlag, New York- Berlin, 1974.
[1] Witt, Ernst Über ein Gegenbeispiel zum Normensatz, Math. Zeit. 39 (1934), 462–467.
[2] Witt, Ernst Schiefkörper über diskret bewerteten Körpern, J. reine angew. Math. 176 (1936), 153–156.
[1] Zariski, Oscar and Samuel, Pierre Commutative Algebra, vol. II, Van Nostrand, Princeton, 1960.


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