Book Review
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MathSciNet review:
3077140
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Book Information:
Author:
T. Y. Lam
Title:
Introduction to quadratic forms over fields
Additional book information:
Graduate Studies in Mathematics, vol. 67, American Mathematical Society,
Providence, RI,
2005,
xxi+550 pp.,
ISBN 978-0-8218-1095-8
E. Bayer-Fluckiger and H. W. Lenstra Jr., Forms in odd degree extensions and self-dual normal bases, Amer. J. Math. 112 (1990), no. 3, 359–373. MR 1055648, DOI 10.2307/2374746
E. Bayer-Fluckiger and R. Parimala, Galois cohomology of the classical groups over fields of cohomological dimension $\leq 2$, Invent. Math. 122 (1995), no. 2, 195–229. MR 1358975, DOI 10.1007/BF01231443
J.-L. Colliot-Thélène, P. Gille, and R. Parimala, Arithmetic of linear algebraic groups over 2-dimensional geometric fields, Duke Math. J. 121 (2004), no. 2, 285–341. MR 2034644, DOI 10.1215/S0012-7094-04-12124-4
Philippe Gille, Cohomologie galoisienne des groupes quasi-déployés sur des corps de dimension cohomologique $\leq 2$, Compositio Math. 125 (2001), no. 3, 283–325 (French, with English summary). MR 1818983, DOI 10.1023/A:1002473132282
Oleg T. Izhboldin, Fields of $u$-invariant $9$, Ann. of Math. (2) 154 (2001), no. 3, 529–587. MR 1884616, DOI 10.2307/3062141
Bruno Kahn, La conjecture de Milnor (d’après V. Voevodsky), Astérisque 245 (1997), Exp. No. 834, 5, 379–418 (French, with French summary). Séminaire Bourbaki, Vol. 1996/97. MR 1627119
Bruno Kahn, Formes quadratiques et cycles algébriques (d’après Rost, Voevodsky, Vishik, Karpenko et al.), Astérisque 307 (2006), Exp. No. 941, vii, 113–163 (French, with French summary). Séminaire Bourbaki. Vol. 2004/2005. MR 2296417
Max-Albert Knus, Quadratic and Hermitian forms over rings, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 294, Springer-Verlag, Berlin, 1991. With a foreword by I. Bertuccioni. MR 1096299, DOI 10.1007/978-3-642-75401-2
9. M. Knus, A. Merkurjev, M. Rost, J-P. Tignol, The Book of Involutions, AMS Colloquium Publications, Vol 44 (1998).
T. Y. Lam, The algebraic theory of quadratic forms, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Reading, Mass., 1973. MR 0396410
A. S. Merkur′ev, Kaplansky’s conjecture in the theory of quadratic forms, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 175 (1989), no. Kol′tsa i Moduli. 3, 75–89, 163–164 (Russian); English transl., J. Soviet Math. 57 (1991), no. 6, 3489–3497. MR 1047239, DOI 10.1007/BF01100118
John Milnor, Algebraic $K$-theory and quadratic forms, Invent. Math. 9 (1969/70), 318–344. MR 260844, DOI 10.1007/BF01425486
Fabien Morel, Milnor’s conjecture on quadratic forms and mod 2 motivic complexes, Rend. Sem. Mat. Univ. Padova 114 (2005), 63–101 (2006). MR 2207862
14. D. Orlov, A. Vishik, V. Voedovsky, An exact sequence for Milnor's 
-theory with applications to quadratic forms, preprint (2001), arxiv.org/abs/math/0101023
Albrecht Pfister, Quadratic forms with applications to algebraic geometry and topology, London Mathematical Society Lecture Note Series, vol. 217, Cambridge University Press, Cambridge, 1995. MR 1366652, DOI 10.1017/CBO9780511526077
A. Pfister, On the Milnor conjectures: history, influence, applications, Jahresber. Deutsch. Math.-Verein. 102 (2000), no. 1, 15–41. MR 1769021
Winfried Scharlau, Quadratic and Hermitian forms, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 270, Springer-Verlag, Berlin, 1985. MR 770063, DOI 10.1007/978-3-642-69971-9
Jean-Pierre Serre, Cohomologie galoisienne des groupes algébriques linéaires, Colloq. Théorie des Groupes Algébriques (Bruxelles, 1962) Librairie Universitaire, Louvain; Gauthier-Villars, Paris, 1962, pp. 53–68 (French). MR 0186719
Jean-Pierre Serre, Cohomologie galoisienne, 5th ed., Lecture Notes in Mathematics, vol. 5, Springer-Verlag, Berlin, 1994 (French). MR 1324577, DOI 10.1007/BFb0108758
Jean-Pierre Serre, Cohomologie galoisienne, 5th ed., Lecture Notes in Mathematics, vol. 5, Springer-Verlag, Berlin, 1994 (French). MR 1324577, DOI 10.1007/BFb0108758
Robert Steinberg, Regular elements of semisimple algebraic groups, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 49–80. MR 180554
22. A. Vishik, Fields of u-invariant 
, preprint (2006).
Vladimir Voevodsky, Reduced power operations in motivic cohomology, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 1–57. MR 2031198, DOI 10.1007/s10240-003-0009-z
Vladimir Voevodsky, Reduced power operations in motivic cohomology, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 1–57. MR 2031198, DOI 10.1007/s10240-003-0009-z
André Weil, Algebras with involutions and the classical groups, J. Indian Math. Soc. (N.S.) 24 (1960), 589–623 (1961). MR 136682
26. E. Witt, Theorie der quadratischen Formen in beliebigen Körpern, J. Reine Angew. Math. 176 (1937), 31-44 (= Collected Papers, 1).
- 1.
- E. Bayer-Fluckiger, H.W. Lenstra, Jr., Forms in odd degree extensions and self-dual normal bases, Amer. J. Math. 112 (1990), 359-373. MR 1055648
- 2.
- E. Bayer-Fluckiger, R. Parimala, Galois cohomology of linear algebraic groups over fields of cohomological dimension
, Invent. Math., 122 (1995) MR 1358975
- 3.
- J.-L. Colliot-Thélène, Ph. Gille, R. Parimala, Arithmetic of linear algebraic groups over two-dimensional geometric fields, Duke Math. J. 121 (2004), 285-321. 195-229. MR 2034644
- 4.
- Ph. Gille, Cohomologie galoisienne des groupes quasi-déployes sur des corps de dimension cohomologique
, Comp. Math. 125 (2001), 283-325. MR 1818983
- 5.
- O. Izhboldin, Fields of
-invariant 
, Ann. of Math. 154 (2001), 529-587. MR 1884616
- 6.
- B. Kahn, La conjecture de Milnor, d'après Voevodsky, Séminaire Bourbaki (1996/97), Exp. 834 (juin 1997), Astérisque 245 (1997), 379-418. MR 1627119
- 7.
- B. Kahn, Formes quadratiques et cycles algébriques, d'après Rost, Voevodsky, Visik, Karpenko, Séminaire Bourbaki (2004/05), Exp. 941 (novembre 2004), Astérisque 245 (1997), 379-418. MR 2296417
- 8.
- M. Knus, Quadratic and hermitian forms over rings, Grundlehren der Math. Wiss. 294 Springer-Verlag (1991). MR 1096299
- 9.
- M. Knus, A. Merkurjev, M. Rost, J-P. Tignol, The Book of Involutions, AMS Colloquium Publications, Vol 44 (1998).
- 10.
- T.Y. Lam, The Algebraic Theory of Quadratic Forms, Benjamin (1973) (revised printing 1980). MR 0396410
- 11.
- A.S. Merkurjev, Kaplansky's conjecture in the theory of quadratic forms (in Russian), Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov (LOMI) 175 (1989), Kolsta i Moduli 3, 75-89, 163-164 (English translation : J. Soviet Math. 57 (1991), 3489-3497). MR 1047239
- 12.
- J. Milnor, Algebraic
-theory and quadratic forms, Invent. Math. 9 (1970), 318-344. MR 0260844
- 13.
- F. Morel, Milnor's conjecture on quadratic forms and mod
motivic complexes, Rend. Sem. Mat. Univ. Padova, 114 (2005), 63-101. MR 2207862
- 14.
- D. Orlov, A. Vishik, V. Voedovsky, An exact sequence for Milnor's -theory with applications to quadratic forms, preprint (2001), arxiv.org/abs/math/0101023
- 15.
- A. Pfister, Quadratic forms with applications to algebraic geometry and topology, LMS Lecture Note Series 217, Cambridge University Press (1995). MR 1366652
- 16.
- A. Pfister, On the Milnor Conjectures, History, Influence, Applications, Jahresbericht DMV 102 (2000), 15-41. MR 1769021
- 17.
- W. Scharlau, Quadratic and Hermitian Forms, Grundlehren der Math. Wiss. 270 Springer-Verlag (1985). MR 0770063
- 18.
- J-P. Serre, Cohomologie galoisienne des groupes algébriques linéaires, Colloque de Bruxelles, 1962, 53-67 (=Collected Papers, 53). MR 0186719
- 19.
- J-P. Serre, Cohomologie galoisienne, Cinquième édition, Lecture Notes in Mathematics 5, Springer-Verlag (1964, 1994) MR 1324577
- 20.
- J-P. Serre, Cohomologie galoisienne : progrès et problèmes, Séminaire Bourbaki 1993-1994, exposé 783. MR 1324577
- 21.
- R. Steinberg, Regular elements of semisimple algebraic groups, Pulb. Math. IHES 25 (1965), 49-80. MR 0180554
- 22.
- A. Vishik, Fields of u-invariant , preprint (2006).
- 23.
- V. Voevodsky, Reduced power operations in motivic cohomology, Publ. Math. IHES 98 (2003), 1-57. MR 2031198
- 24.
- V. Voevodsky, Motivic cohomology with
-coefficients, Publ. Math. IHES 98 (2003), 59-104. MR 2031198
- 25.
- A. Weil, Algebras with involutions and the classical groups, J. Indian Math. Soc. (N.S.) 24 (1960) (= Collected Papers, 1960 b). MR 0136682
- 26.
- E. Witt, Theorie der quadratischen Formen in beliebigen Körpern, J. Reine Angew. Math. 176 (1937), 31-44 (= Collected Papers, 1).
Review Information:
Reviewer:
Eva Bayer-Fluckiger
Affiliation:
Ecole Polytechnique Fédérale, Lausanne
Email:
eva.baver@epfl.ch
Journal:
Bull. Amer. Math. Soc.
45 (2008), 479-484
DOI:
https://doi.org/10.1090/S0273-0979-08-01200-7
Published electronically:
April 21, 2008
Review copyright:
© Copyright 2008
American Mathematical Society
