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Book Information:
Author:
Ido Efrat
Title:
Valuations, orderings and Milnor

-theory
Additional book information:
Mathematical Surveys and Monographs, vol. 124,
American Mathematical Society, Providence, RI,
2006,
xiv+288 pp.,
ISBN 978-0-8218-4041-2,
US$60.00$
Jón Kr. Arason, Richard Elman, and Bill Jacob, Rigid elements, valuations, and realization of Witt rings, J. Algebra 110 (1987), no. 2, 449–467. MR 910395, DOI 10.1016/0021-8693(87)90057-3
Eberhard Becker and Ludwig Bröcker, On the description of the reduced Witt ring, J. Algebra 52 (1978), no. 2, 328–346. MR 506029, DOI 10.1016/0021-8693(78)90243-0
Eberhard Becker and Alex Rosenberg, Reduced forms and reduced Witt rings of higher level, J. Algebra 92 (1985), no. 2, 477–503. MR 778463, DOI 10.1016/0021-8693(85)90135-8
Ludwig Bröcker, Zur Theorie der quadratischen Formen über formal reellen Körpern, Math. Ann. 210 (1974), 233–256 (German). MR 354549, DOI 10.1007/BF01350587
Ludwig Bröcker, Characterization of fans and hereditarily Pythagorean fields, Math. Z. 151 (1976), no. 2, 149–163. MR 422233, DOI 10.1007/BF01213992
Thomas C. Craven, Characterizing reduced Witt rings of fields, J. Algebra 53 (1978), no. 1, 68–77. MR 480332, DOI 10.1016/0021-8693(78)90205-3
[E] I. Efrat, Orderings, valuations and free products of Galois groups, Sém. Struct. Alg. Ordonnées 54, Univ. Paris 7 (1995).
Yoon Sung Hwang and Bill Jacob, Brauer group analogues of results relating the Witt ring to valuations and Galois theory, Canad. J. Math. 47 (1995), no. 3, 527–543. MR 1346152, DOI 10.4153/CJM-1995-029-4
Bill Jacob and Roger Ware, A recursive description of the maximal pro-$2$ Galois group via Witt rings, Math. Z. 200 (1989), no. 3, 379–396. MR 978598, DOI 10.1007/BF01215654
Bill Jacob and Roger Ware, Realizing dyadic factors of elementary type Witt rings and pro-$2$ Galois groups, Math. Z. 208 (1991), no. 2, 193–208. MR 1128705, DOI 10.1007/BF02571520
Jochen Koenigsmann, From $p$-rigid elements to valuations (with a Galois-characterization of $p$-adic fields), J. Reine Angew. Math. 465 (1995), 165–182. With an appendix by Florian Pop. MR 1344135, DOI 10.1515/crll.1995.465.165
Mieczysław Kula, Fields with prescribed quadratic form schemes, Math. Z. 167 (1979), no. 3, 201–212. MR 539104, DOI 10.1007/BF01174801
Tsit Yuen Lam, Orderings, valuations and quadratic forms, CBMS Regional Conference Series in Mathematics, vol. 52, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1983. MR 714331, DOI 10.1090/cbms/052
M. Marshall, The elementary type conjecture in quadratic form theory, Algebraic and arithmetic theory of quadratic forms, Contemp. Math., vol. 344, Amer. Math. Soc., Providence, RI, 2004, pp. 275–293. MR 2060204, DOI 10.1090/conm/344/06224
M. Marshall, Real reduced multirings and multifields, J. Pure Appl. Algebra 205 (2006), no. 2, 452–468. MR 2203627, DOI 10.1016/j.jpaa.2005.07.011
A. S. Merkur′ev and A. A. Suslin, $K$-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 1011–1046, 1135–1136 (Russian). MR 675529
John Milnor, Algebraic $K$-theory and quadratic forms, Invent. Math. 9 (1969/70), 318–344. MR 260844, DOI 10.1007/BF01425486
Ján Mináč and Tara L. Smith, Decomposition of Witt rings and Galois groups, Canad. J. Math. 47 (1995), no. 6, 1274–1289. MR 1370518, DOI 10.4153/CJM-1995-065-0
Albrecht Pfister, Quadratische Formen in beliebigen Körpern, Invent. Math. 1 (1966), 116–132 (German). MR 200270, DOI 10.1007/BF01389724
Victoria Powers, Characterizing reduced Witt rings of higher level, Pacific J. Math. 128 (1987), no. 2, 333–347. MR 888522
Alexander Prestel, Lectures on formally real fields, Lecture Notes in Mathematics, vol. 1093, Springer-Verlag, Berlin, 1984. MR 769847, DOI 10.1007/BFb0101548
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
[V2] -, On motivic cohomology with
-coefficients, preprint.
[W] E. Witt, Theorie der Quadratischen Formen in beliebigen Körpern, J. Reine Angew. Math. 176 (1937), 31-44.
- [A-E-J]
- J. Arason, R. Elman, B. Jacob, Rigid elements, valuations, and realization of Witt rings, J. Algebra 110 (1987), 449-467. MR 0910395
- [B-B]
- E. Becker, L. Bröcker, On the description of the reduced Witt ring, J. Algebra 52 (1978), 328-346. MR 0506029
- [B-R]
- E. Becker, A. Rosenberg, Reduced forms and reduced Witt rings of higher level, J. Algebra 92 (1985), 477-503. MR 0778463
- [B1]
- L. Bröcker, Zur Theorie der quadratischen Formen über formal reellen Körpern, Math. Ann. 210 (1974), 233-256. MR 0354549
- [B2]
- -, Characterization of fans and hereditarily Pythagorean fields, Math. Z. 151 (1976), 149-163. MR 0422233
- [C]
- T. Craven, Characterizing reduced Witt rings of fields, J. Algebra 53 (1978), 74-96. MR 0480332
- [E]
- I. Efrat, Orderings, valuations and free products of Galois groups, Sém. Struct. Alg. Ordonnées 54, Univ. Paris 7 (1995).
- [H-J]
- Y.S. Hwang, B. Jacob, Brauer group analogues of results relating the Witt ring to valuations and Galois theory, Canad. J. Math. 47 (1995), 527-543. MR 1346152
- [J-W1]
- B. Jacob, R. Ware, A recursive description of the maximal pro-
Galois group via Witt rings, Math. Z. 200 (1989), 379-396. MR 0978598
- [J-W2]
- -, Realizing dyadic factors of elementary type Witt rings and pro-2 Galois groups, Math. Z. 208 (1991), 193-208. MR 1128705
- [Ko]
- J. Koenigsmann, From
-rigid elements to valuations (with a Galois-characterization of
-adic fields), J. Reine Angew. Math. 465 (1995), 165-182. MR 1344135
- [Ku]
- M. Kula, Fields with prescribed quadratic form schemes, Math. Z. 167 (1979), 201-212. MR 0539104
- [L]
- T.-Y. Lam, Orderings, valuations and quadratic forms, CBMS 52, Amer. Math. Soc., 1983. MR 0714331
- [M1]
- M. Marshall, The elementary type conjecture in quadratic form theory, Cont. Math. 344 (2004), 275-293. MR 2060204
- [M2]
- -, Real reduced multirings and multifields, J. Pure and Applied Algebra 205 (2006), 452-468. MR 2203627
- [M-S]
- A.S. Merkurjev, A.A. Suslin,
-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 1011-1046 (Russian); Math. USSR Izv. 21 (1983), 307-340 (English translation). MR 0675529
- [Mi]
- J. Milnor, Algebraic K-theory and quadratic forms, Invent. Math. 9 (1970), 318-344. MR 0260844
- [Mi-S]
- J. Minác, T. Smith, Decomposition of Witt rings and Galois groups, Canad. J. Math. 47 (1995), 1274-1289. MR 1370518
- [Pf]
- A. Pfister, Quadratische Formen in beliebigen Körpern, Invent. Math. (1966), 116-132. MR 0200270
- [Po]
- V. Powers, Characterizing reduced Witt rings of higher level, Pac. J. Math. 128 (1987), 333-347. MR 0888522
- [Pr]
- A. Prestel, Lectures on formally real fields, IMPA Lecture Notes 22, Rio de Janeiro, 1975, Lecture Notes in Math. 1093, Springer, 1984. MR 0769847
- [V1]
- V. Voevodsky, Motivic cohomology with
-coefficients, Publ. Math. IHES 98 (2003), 59-104. MR 2031198
- [V2]
- -, On motivic cohomology with
-coefficients, preprint.
- [W]
- E. Witt, Theorie der Quadratischen Formen in beliebigen Körpern, J. Reine Angew. Math. 176 (1937), 31-44.
Review Information:
Reviewer:
Murray Marshall
Affiliation:
University of Saskatchewan
Email:
marshall@math.usask.ca
Journal:
Bull. Amer. Math. Soc.
45 (2008), 439-444
Published electronically:
August 3, 2007
Review copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.