×

Quadratic forms. (English) Zbl 1213.11092

Hazewinkel, M. (ed.), Handbook of algebra. Volume 6. Amsterdam: Elsevier/North-Holland (ISBN 978-0-444-53257-2/hbk). Handbook of Algebra 6, 35-80 (2009).
This is a valuable survey on quadratic forms. After the introductory material the author discusses briefly the Milnor Conjecture and its solution by Voevodsky. Next he presents the classification results for quadratic forms and Witt rings, considering the classification of Witt rings over a field \(F\) to be the ultimate goal of the theory, and reports on the results over the fields where the theory is complete. Selected applications of function fields of quadratic forms are discussed also, sums of squares, Witt rings of rings, abstract Witt rings. The paper ends with a historical section.
For the entire collection see [Zbl 1182.00007].

MSC:

11E81 Algebraic theory of quadratic forms; Witt groups and rings
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11E25 Sums of squares and representations by other particular quadratic forms
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adem, A.; Gao, W.; Karagueuzian, D. B.; Mináč, J., Field theory and the cohomology of some Galois groups, J. Algebra, 235, 608-635 (2001) · Zbl 0973.12005
[2] Arason, J. K., Cohomologische Invarianten quadratischer Formen, J. Algebra, 36, 448-491 (1975) · Zbl 0314.12104
[3] Arason, J. K., Der Wittring projektiver Räume, Math. Ann., 253, 205-212 (1980) · Zbl 0431.10011
[4] Quadratic and Hermitian forms (1983), Hamilton: Hamilton Ontario
[5] Arason, J. K.; Pfister, A., Beweis des Krullschen Durchschnittsatzes für den Wittring, Invent. Math., 12, 173-176 (1971) · Zbl 0212.37302
[6] Arason, J. K.; Elman, R., Powers of the fundamental ideal in the Witt ring, J. Algebra, 239, 1, 150-160 (2001) · Zbl 0990.11021
[7] Arason, J. K.; Elman, R.; Jacob, B., On the Witt ring of elliptic curves, Proc. Symp. Pure Math., 58, Part II, 1-25 (1995) · Zbl 0823.11017
[8] Artin, E., Über die Zerlegung definiter Funktionen in Quadrate, Abh. Math. Sem. Univ. Hamburg, 5, 100-115 (1927) · JFM 52.0122.01
[9] Artin, E.; Schreier, O., Algebraische Konstruktion reeller Körper, Abh. Math. Sem. Univ. Hamburg, 5, 85-99 (1927) · JFM 52.0120.05
[10] Baeza, R., Lecture Notes in Mathematics, (Quadratic forms over semilocal rings, vol. 655 (1978), Springer Verlag: Springer Verlag Berlin)
[11] Contemporary Mathematics, (Baeza, R.; Hsia, J. S.; Jacob, B.; Prestel, A., Algebraic and Arithmetic Theory of Quadratic forms, vol. 344 (2004), American Mathematical Society Providence: American Mathematical Society Providence RI) · Zbl 0142.24201
[12] Baeza, R.; Moresi, R., On the Witt-equivalence of fields of characteristic 2, J. Algebra, 92, 446-453 (1985) · Zbl 0553.10016
[13] Balmer, P., Triangular Witt groups, I: The 12-term localization exact sequence, K-theory, 19, 311-363 (2000) · Zbl 0953.18003
[14] Balmer, P., Triangular Witt groups, II: From usual to derived, Math. Zeit., 236, 351-382 (2001) · Zbl 1004.18010
[15] Balmer, P., (Handbook of K-theory, vol. 2: Witt Groups (2005), Springer Verlag), 539-576
[16] Bayer-Fluckiger, E.; Parimala, R., Galois cohomology of the classical groups over fields of coho-mological dimension ≤2, Invent. Math., 122, 195-229 (1995) · Zbl 0851.11024
[17] Becker, E., Hereditarily Pythagorean Fields and Orderings of Higher Level, (Monografias de Matemática, vol. 29 (1978), Instituto de Matemática Aplicada: Instituto de Matemática Aplicada Rio de Janeiro) · Zbl 0509.12020
[18] E. Becker, Lecture Notes in Mathematics, vol. 959: The real holomorphy ring and sums of 2nth powers; E. Becker, Lecture Notes in Mathematics, vol. 959: The real holomorphy ring and sums of 2nth powers
[19] Becker, E.; Bröcker, L., On the description of the reduced Witt ring, J. Algebra, 52, 328-346 (1978) · Zbl 0396.10012
[20] Becker, E.; Harman, J.; Rosenberg, A., Signatures of fields and extwnsion theory, J. ReineAngew Math., 330, 53-75 (1982) · Zbl 0466.12007
[21] Becker, E.; Rosenberg, A., Reduced forms and reduced Witt rings of higher level, J. Algebra, 92, 477-503 (1985) · Zbl 0555.10009
[22] Brauer, R., Emil Artin, Bull. Am. Math. Soc., 73, 27-43 (1967) · Zbl 0147.00509
[23] Bröcker, L., Über dieAnzahl derAnordnungen eines kommutativen Körpers, Arch. Math., 29, 458-464 (1977) · Zbl 0368.12014
[24] Carpenter, J. P., Finiteness theorems for forms over global fields, Math. Zeit., 209, 153-166 (1992) · Zbl 0724.11021
[25] Carson, A. B.; Marshall, M. A., Decomposition of Witt rings, Can. J. Math., 34, 1276-1302 (1982) · Zbl 0516.10014
[26] Cassels, J. W.S., On the representation of rational functions as sums of squares, Acta Arith., 9, 79-82 (1964) · Zbl 0131.25001
[27] Cassels, J. W.S.; Ellison, W. J.; Pfister, A., On sums of squares and on elliptic curves over function fields, J. Number Theory, 3, 125-149 (1971) · Zbl 0217.04302
[28] Choi, M. D.; Dai, Z. D.; Lam, T. Y.; Reznick, B., The Pythagoras number of some affine algebras and local algebras, J. Reine Angew. Math., 336, 45-82 (1982) · Zbl 0499.12018
[29] Choi, M. D.; Lam, T. Y.; Prestel, A.; Reznick, B., Sums of \(2m\) th powers of rational functions in one variable over real closed fields, Math. Zeit., 221, 93-112 (1996) · Zbl 0859.12002
[30] Ciemała, M.; Szymiczek, K., On natural homomorphisms of Witt rings, Proc. Am. Math. Soc., 133, 2519-2523 (2005) · Zbl 1071.11020
[31] Ciemała, M.; Szymiczek, K., On injectivity of natural homomorphisms of Witt rings, Ann. Math Siles., 21, 15-30 (2007) · Zbl 1241.11046
[32] Colliot-Thélène, J.-L.; Jannsen, U., Sommes de carrés dans les corps de fonctions, C.R. Acad. Sci Paris, 312, 759-762 (1991) · Zbl 0743.11020
[33] Colliot-Thélène, J.-L.; Sujatha, R., The unramified Witt group of real anisotropic quadrics, Proc Symp. Pure Math., 58, Part II, 127-147 (1995) · Zbl 0827.11022
[34] Conner, P. E.; Perlis, R., A Survey of Trace Forms of Algebraic Number Fields (1984), World Scientific: World Scientific Singapore · Zbl 0551.10017
[35] Conner, P. E.; Perlis, R.; Szymiczek, K., Wild sets and 2-ranks of class groups, Acta Arith., 79, 83-91 (1997) · Zbl 0880.11039
[36] Conner, P. E.; Yui, N., The additive characters of the Witt ring of an algebraic number field, Can. J Math., 40, 546-588 (1988) · Zbl 0647.12006
[37] Conway, J. H.; Sloane, N. J.A., Grundlehren der mathematischen Wissenschaften, (Sphere Packings, Lattices, and Groups, vol. 209 (1988), Springer-Verlag: Springer-Verlag New York)
[38] Cordes, C. M., The Witt group and equivalence of fields with respect to quadratic forms, J. Algebra, 26, 400-421 (1973) · Zbl 0288.12101
[39] Cordes, C. M., Quadratic forms over nonformally real fields with a finite number of quaternion algebras, Pacific J. Math., 63, 357-365 (1976) · Zbl 0335.10024
[40] Cordes, C. M., Quadratic forms over fields with four quaternion algebras, Acta Arith., 41, 55-70 (1982) · Zbl 0416.10014
[41] Craven, T. C.; Rosenberg, A.; Ware, R., The map of the Witt ring of a domain into the Witt ring of its field of fractions, Proc. Am. Math. Soc., 51, 25-30 (1975) · Zbl 0313.13025
[42] Craven, T. C., The Boolean space oforderingsof a field, Trans. Am. Math. Soc., 209, 225-235 (1975) · Zbl 0315.12106
[43] Craven, T. C., Characterizing reduced Witt rings of fields, J. Algebra, 53, 68-77 (1978) · Zbl 0376.12008
[44] Craven, T. C., Characterizing reduced Witt rings, II. Pacific J. Math., 80, 341-349 (1979) · Zbl 0376.12009
[45] Craven, T. C., On the topological space of all orderings of a skew field, Arch. Math., 36, 234-238 (1981) · Zbl 0439.12013
[46] Craven, T. C., Witt rings and orderings of skew fields, J. Algebra, 77, 74-96 (1982) · Zbl 0493.10026
[47] Czogała, A., On reciprocity equivalence of quadratic number fields, Acta Arith., 58, 27-46 (1991) · Zbl 0733.11012
[48] Czogała, A., On integral Witt equivalence of algberaic number fields, Acta Math. Inf. Univ. Ostrav, 4, 7-21 (1996) · Zbl 0870.11022
[49] Czogała, A., Higher degree tame Hilbert-symbol equivalence of number fields, Abh. Math. Sem Univ. Hamburg, 69, 175-185 (1999) · Zbl 0968.11038
[50] Czogała, A., Witt rings of Hasse domains of global fields, J. Algebra, 244, 604-630 (2001) · Zbl 0984.11016
[51] Czogała, A., Hilbert-symbol equivalence of global function fields, Math. Slovaca, 51, 393-401 (2001) · Zbl 0987.11072
[52] Czogała, A.; Sładek, A., Higher degree Hilbert-symbol equivalence of number fields, I. Tatra Mount Math. Publ., 11, 77-88 (1997) · Zbl 0978.11058
[53] Czogała, A.; Sładek, A., Higher degree Hilbert-symbol equivalence of number fields, II. J. Number Theory, 72, 363-376 (1998) · Zbl 0922.11096
[54] Czogała, A.; Rothkegel, B., Witt equivalence of semilocal dedekind domains in global fields, Abh Math. Sem. Univ. Hamburg, 77, 1-24 (2007) · Zbl 1143.11012
[55] Czogała, A.; Sładek, A., Witt rings of global fields, Commun. Algebra, 31, 3195-3205 (2003) · Zbl 1069.11012
[56] Dai, Z. D.; Lam, T. Y.; Peng, C. K., Levels in algebra and topology, Bull. Am. Math. Soc., 3, 845-848 (1980) · Zbl 0435.10016
[57] Dai, Z. D.; Lam, T. Y., Levels in algebra and topology, Comment. Math. Helv., 59, 376-424 (1984) · Zbl 0546.10017
[58] DeMeyer, F. R.; Harrison, D. K.; Miranda, R., Quadratic forms over Q and Galois extensions of commutative rings, Mem. Am. Math. Soc., 77, 394, 1-63 (1989) · Zbl 0668.10028
[59] Dickmann, M. A.; Miraglia, F., Special groups: Boolean-theoretic methods in the theory of quadratic forms, Mem. Am. Math. Soc., 145, 689 (2000) · Zbl 1052.11027
[60] Dickmann, M. A.; Miraglia, F., Algebraic \(K\)-theory of special groups, J. Pure Appl. Algebra, 204, 195-234 (2006) · Zbl 1105.19004
[61] Dickson, L. E., On quadratic forms in a general field, Bull. Am. Math. Soc., 14, 108-115 (1907) · JFM 38.0182.02
[62] Dress, A. W.M., The weak local global principle in algebraic \(K\)-theory, Commun. Algebra, 3, 7, 615-661 (1975) · Zbl 0313.18011
[63] Contemporary Mathematics, (Dubois, D. W.; Recio, T., Ordered Fields and Real Algebraic Geometry, vol. 8 (1982), American Mathematical Society: American Mathematical Society Providence, RI), 1-360
[64] Dugger, D.; Isaksen, D., The Hopf condition for bilinear forms over arbitrary fields, Ann. Math., 165, 943-964 (2007) · Zbl 1125.11026
[65] Elman, R.; Karpenko, N.; Merkurjev, A., AMS Colloquium Publications, (The Algebraic and Geometric Theory of Quadratic Forms, vol. 56 (2008), AMS: AMS Providence, RI)
[66] Elman, R.; Lam, T. Y., Quadratic forms and the \(u\)-invariant, I. Math. Zeit., 131, 283-304 (1973) · Zbl 0244.10020
[67] Elman, R.; Lam, T. Y., Quadratic forms and the \(u\)-invariant, II. Invent. Math., 21, 125-137 (1973) · Zbl 0267.10029
[68] Elman, R.; Lam, T. Y., Classification theorems for quadratic forms over fields, Comment. Math. Helv, 49, 373-381 (1974) · Zbl 0297.15024
[69] Elman, R.; Lam, T. Y.; Wadsworth, A. R., Amenable fields and Pfister extensions, Queen’s Papers in Pure and Appl. Math., 46, 445-492 (1976)
[70] Elman, R.; Lam, T. Y.; Wadsworth, A. R., Function fields of Pfister forms, Invent. Math., 51, 61-75 (1979) · Zbl 0395.10028
[71] Elman, R.; Lam, T. Y.; Wadsworth, A. R., Quadratic forms under multiquadratic extensions, Indag Math., 42, 131-145 (1980) · Zbl 0425.10021
[72] Elman, R.; Lam, T. Y.; Tignol, J.-P.; Wadsworth, A. R., Witt rings and Brauer groups under multi-quadratic extensions, I. Am. J. Math., 105, 1119-1170 (1983) · Zbl 0492.10014
[73] Epkenhans, M., Trace forms of normal extensions of local fields, Linear and Multilinear Algebra, 24, 103-116 (1989) · Zbl 0702.11019
[74] Epkenhans, M., Trace forms of normal extensionsof algebraic numberfields, Linear and Multilinear Algebra, 25, 309-320 (1989) · Zbl 0688.10019
[75] Epkenhans, M., On the ramification set of a positive quadratic form over an algebraic number field, Acta Arith., 66, 133-145 (1994) · Zbl 0793.11013
[76] Estes, D. R.; Hurrelbrink, J.; Perlis, R., Total positivity and algebraic Witt classes, Comment. Math Helvetici, 60, 284-290 (1985) · Zbl 0589.10021
[77] Fernando, J.; Ruiz, J.; Scheiderer, C., Sums of squares in real rings, Trans. Am. Math. Soc., 356, 2663-2684 (2004) · Zbl 1080.14071
[78] Fitzgerald, R. W., Primary ideals in Witt rings, J. Algebra, 96, 368-385 (1985) · Zbl 0574.10028
[79] Fitzgerald, R. W., Ideal class groups of Witt rings, J. Algebra, 124, 506-520 (1989) · Zbl 0682.10016
[80] Fitzgerald, R. W., Picard groups of Witt rings, Math. Zeit., 296, 303-319 (1991) · Zbl 0702.11024
[81] Fitzgerald, R. W., Witt rings under odd degree extensions, Pacific J. Math., 158, 121-143 (1993) · Zbl 0790.11034
[82] Gao, W.; Mináč, J., Milnor’s conjecture and Galois theory, I. Fields Inst. Commun., 16, 95-110 (1997) · Zbl 0883.12003
[83] Gerstein, L. J., (Basic Quadratic Forms, Graduate Studies in Math., vol. 90 (2008), Am. Math. Soc.: Am. Math. Soc. Providence, RI) · Zbl 1147.11002
[84] Harrison, D. K., Witt Rings (1970), University of Kentucky: University of Kentucky Lexington, Kentucky, Notes by Joel Cunningham.
[85] Harrison, D. K.; Pareigis, B., Witt rings of higher degree forms, Commun. Algebra, 16, 1275-1313 (1988) · Zbl 0645.10022
[86] Hasse, H., Über die Darstellbarkeit von Zahlen durch quadratische Formen im Körper der rationalen Zahlen, J. Reine Angew. Math., 152, 129-148 (1923) · JFM 49.0102.01
[87] Hasse, H., Über die Äquivalenz quadratischer Formen im Körper der rationalen Zahlen, J. Reine Angew. Math., 152, 205-224 (1923) · JFM 49.0102.02
[88] Hasse, H., Symmetrische Matrizen im Körper der rationalen Zahlen, J. Reine Angew. Math., 153, 12-43 (1924) · JFM 49.0104.01
[89] Hasse, H., Darstellbarkeit von Zahlen durch quadratische Formenin einem beliebigen algebraischen Zahlkörper, J. Reine Angew. Math., 153, 113-130 (1924) · JFM 49.0114.01
[90] Hasse, H., Äquivalenz quadratischer Formenineinem beliebigen algebraischen Zahlkörper, J. Reine Angew. Math., 153, 158-162 (1924) · JFM 50.0104.03
[91] Hilbert, D., Über die Theorie des relativ quadratischen Zahlkörpers, Math. Ann., 51, 1-127 (1899) · JFM 29.0169.02
[92] Hoffmann, D. W., Isotropy of quadratic forms over the function field of a quadric, Math. Zeit., 220, 461-476 (1995) · Zbl 0840.11017
[93] Hoffmann, D. W., Pythagoras numbers of fields, J. Am. Math. Soc., 12, 839-848 (1999) · Zbl 0921.11018
[94] Hoffmann, D. W., Isotropy of quadratic forms and field invariants, Contemp. Math., 272, 73-102 (2000) · Zbl 1025.11006
[95] Hoffmann, D. W., Levels of quaternion algebras, Arch. Math. (Basel), 90, 401-411 (2008) · Zbl 1195.11047
[96] Hornix, E. A.M., Formally real fields with prescribed invariants in the theory of quadratic forms, Indag. Math. (N.S.), 1, 2, 65-78 (1991) · Zbl 0734.11028
[97] Hurrelbrink, J., Annihilating polynomials for group rings and Witt rings, Can. Math. Bull, 32, 412-416 (1989) · Zbl 0636.10019
[98] Hurrelbrink, J.; Karpenko, N. A.; Rehman, U., The minimal height of quadratic forms of given dimension, Arch. Math., 87, 522-529 (2006) · Zbl 1109.11023
[99] Izhboldin, O. T., Fields of \(u\)-invariant 9, Ann. Math., 154, 529-587 (2001) · Zbl 0998.11015
[100] Izhboldin, O. T.; Kahn, B.; Karpenko, N. A.; Vishik, A., Lecture Notes in Mathematics, vol. 1835: Geometric Methods in the Algebraic Theory of Quadratic Forms (2004), Springer: Springer Berlin, Heidelberg
[101] Jacob, B., Quadratic forms over dyadic valued fields, I. The graded Witt ring, Pacific J. Math., 126, 21-79 (1987) · Zbl 0606.10013
[102] Jacob, B., Quadratic forms over dyadic valued fields, II. Relative rigidity and Galois cohomology, J. Algebra, 148, 162-202 (1992) · Zbl 0762.11013
[103] Jacob, W. B.; Rost, M., Degree four cohomological invariants for quadratic forms, Invent. Math., 96, 551-570 (1989) · Zbl 0685.10015
[104] Jacob, W. B.; Ware, R., A recursive description of the maximal pro-2 Galois group via Witt rings, Math. Zeit, 200, 379-396 (1989) · Zbl 0663.12018
[105] Jacob, W. B.; Ware, R., Realizing dyadic factors of elementary type Witt rings and pro-2 Galois groups, Math. Zeit., 208, 193-208 (1991) · Zbl 0755.11014
[106] Contemporary Mathematics, (Jacob, W. B.; Lam, T. Y.; Robson, R. O., Recent Advances in Real Algebraic Geometry and Quadratic Forms, vol. 155 (1994), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 0142.24201
[107] (Jacob, W. B.; Rosenberg, A., K-theory and algebraic geometry: Connections with quadratic forms and division algebras, in Proceedings of Symposia in Pure Mathematics (1995), American Mathematical Society: American Mathematical Society Providence, RI)
[108] Kaplansky, I., Quadratic forms, J. Math. Soc. Japan, 5, 200-207 (1953) · Zbl 0051.02902
[109] Kaplansky, I., Fröhlich’s local quadratic forms, J. Reine Angew. Math., 239/240, 74-77 (1969) · Zbl 0238.15011
[110] Karpenko, N., On the first Witt index of quadratic forms, Invent. Math., 153, 455-462 (2003) · Zbl 1032.11016
[111] Karpenko, N., Holes in \(I^n\), Ann. Sci. École Norm. Sup. (4), 37, 6, 973-1002 (2004) · Zbl 1108.11037
[112] Karpenko, N., (A relation between higherWitt indices, Proceedingsofthe St. Petersburg Mathematical Society, vol. XI (2006), American Mathematical Society: American Mathematical Society Providence, RI), 77-86, Amer. Math. Soc. Transl. Ser. 2, 218
[113] Karpenko, N.; Merkurjev, A., Essential dimensions of quadrics, Invent. Math., 153, 361-372 (2003) · Zbl 1032.11015
[114] (Kim, M.-H.; Hsia, J. S.; Kitaoka, Y.; Schulze-Pillot, R., Contemporary Mathematics, Integral quadratic forms and lattices, vol. 249 (1999), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 0931.00029
[115] Knebusch, M., (Grothendieck- und Wittringe von nichtausgearteten symmetrischen Bilinearformen. Sitzungsber. Heidelb. Akad. Wiss. Mat.-naturw. Kl., vol. 3 (1969/1970), Springer Verlag: Springer Verlag Berlin)
[116] Knebusch, M., Runde Formen über semilokalen Ringen, Math. Ann., 193, 21-34 (1971) · Zbl 0217.04701
[117] Knebusch, M., Real closures of commutative rings, I. J. Reine Angew. Math., 274/275, 61-89 (1975) · Zbl 0331.13007
[118] Knebusch, M., Real closures of commutative rings, II. J. Reine Angew. Math., 286/287, 278-313 (1976) · Zbl 0332.13016
[119] Knebusch, M., Generic splitting of quadratic forms, I. Proc. London Math. Soc. (3), 33, 1, 65-93 (1976) · Zbl 0351.15016
[120] Knebusch, M., Symmetric bilinear forms over algebraic varieties, Queen’s Papers in Pure andAppl Math., 46, 103-283 (1977)
[121] Knebusch, M.; Rosenberg, A.; Ware, R., Structure of Witt rings and quotients of Abelian group rings, Am. J. Math., 94, 119-155 (1972) · Zbl 0248.13030
[122] Knebusch, M.; Rosenberg, A.; Ware, R., Grothendieck and Witt rings of hermitian forms over Dedekind rings, Pacific J. Math., 43, 657-673 (1972) · Zbl 0256.15019
[123] Knebusch, M.; Rosenberg, A.; Ware, R., Signatures on semi-local rings, J. Algebra, 26, 208-250 (1973) · Zbl 0273.13016
[124] Knebusch, M.; Rehmann, U., Generic splitting towers and generic splitting preparation of quadratic forms, Contemp. Math., 272, 173-199 (2000) · Zbl 0995.11028
[125] Kneser, M., Quadratische Formen (2002), Springer-Verlag: Springer-Verlag Berlin
[126] Knus, M.-A., Quadratic and Hermitian forms over rings (1991), Springer-Verlag: Springer-Verlag Berlin-Heidelberg-New York · Zbl 0756.11008
[127] Knus, M.-A.; Merkurjev, A.; Rost, M.; Tignol, J.-P., American Mathematical Society Colloqium Publication, vol.44: The book of involutions (1998), American MathematicalSociety: American MathematicalSociety Providence, RI
[128] Knus, M.-A.; Ojanguren, M.; Sridharan, R., Quadratic forms and Azumaya algebras, J. ReineAngew Math., 303/304, 231-248 (1978) · Zbl 0385.16001
[129] Koprowski, P., Witt equivalence of algebraic function fields over real closed fields, Math. Zeit., 242, 323-345 (2002) · Zbl 1067.11020
[130] Koprowski, P., On existence of tame Harrison map, Math. Slovaca, 57, 407-414 (2007) · Zbl 1164.11022
[131] Krüskemper, M., Algebraic number field extensions with prescribed trace forms, J. Number Theory, 40, 120-124 (1992) · Zbl 0762.11014
[132] Kula, M., Fields with prescribed quadratic form schemes, Math. Zeit., 167, 201-212 (1979) · Zbl 0388.10017
[133] Kula, M., Fields with non-trivial Kaplansky’s radical and finite square class number, Acta Arith, 37, 411-418 (1981) · Zbl 0467.10015
[134] Kula, M., Fields and quadratic form schemes, Ann. Math. Siles., 1, 7-22 (1985)
[135] Kula, M., Crystal growth and Witt rings, J. Algebra, 136, 190-196 (1991) · Zbl 0719.11024
[136] Kula, M., Finitely generated Witt rings, Uniw. Ślaski w Katowicach, 1207, 1-52 (1991) · Zbl 0744.11019
[137] Kula, M., Counting Witt rings, J. Algebra, 206, 568-587 (1998) · Zbl 0987.11029
[138] Kula, M.; Marshall, M.; Sładek, A., Direct limits of finite spaces of orderings, Pacific J. Math., 112, 391-406 (1984) · Zbl 0535.10020
[139] Kula, M.; Szczepanik, L.; Szymiczek, K., Quadratic forms over formally real fields with eight square classes, Manuscripta Math., 29, 295-303 (1979) · Zbl 0432.10007
[140] Labute, J.; Lemire, N.; Mináč, J.; Swallow, J., Demuškin groups, Galois modules, and the elementary type conjecture, J. Algebra, 304, 1130-1146 (2006) · Zbl 1168.12003
[141] Lam, T. Y., The theory of ordered fields, (McDonald, B., Ring Theory and Algebra III, Lecture Notes in Pure and Applied Mathematics, vol. 55 (1980), Dekker: Dekker New York), 1-152
[142] Lam, T. Y., Conference Board of Mathematical Sciences Lecture Notes Series, (Orderings, Valuations and Quadratic Forms, vol. 52 (1983), American Mathematical Society: American Mathematical Society Providence, RI)
[143] Lam, T. Y., Graduate Studies in Mathematics, (Introduction to Quadratic Forms over Fields, vol. 67 (2005), American Mathematical Society: American Mathematical Society Providence, RI)
[144] Lam, T. Y., Fields of u-invariant 6 after A. Merkurjev, (Rowen, L., Ring Theory 1989. Proc. Symp and Workshop, Jerusalem/Isr. 1988/89, Israel Mathematical Conference Proceedings 1 (1989), Weizmann Science Press of Israel), 12-30 · Zbl 0683.10018
[145] Leep, D. B., Systems of quadratic forms, J. Reine Angew. Math., 350, 109-116 (1984) · Zbl 0531.10023
[146] Leep, D. B.; Marshall, M. A., Isomorphisms and automorphisms of Witt rings, Can. Math. Bull., 31, 250-256 (1988) · Zbl 0651.10015
[147] Leep, D. B.; Merkurjev, A. S., Growth of the \(u\)-invariant under algebraic extensions, Contemp. Math, 155, 327-332 (1994) · Zbl 0804.11029
[148] Leep, D. B.; Shapiro, D. B.; Wadsworth, A. R., Sums of squares in division algebras, Math. Zeit., 190, 151-162 (1985) · Zbl 0548.16019
[149] Leep, D. B.; Smith, T. L., Witt kernels of triquadratic extensions, Contemp. Math., 344, 249-256 (2004) · Zbl 1143.11314
[150] Lewis, D. W., Levels and sublevels of division algebras, Proc. Roy. Irish Acad. Sect. A, 87, 103-106 (1987) · Zbl 0659.16014
[151] Lewis, D. W., Witt rings as integral rings, Invent. Math., 90, 631-633 (1987) · Zbl 0629.10017
[152] Lewis, D. W., New proofs of the structure theorems for Witt rings, Expos. Math., 7, 83-88 (1989) · Zbl 0665.10014
[153] Lewis, D. W., Sums of squares in Witt rings, J. Pure Appl. Algebra, 69, 67-72 (1990) · Zbl 0716.11019
[154] Lewis, D. W., Units in Witt rings, Commun. Algebra, 18, 3295-3306 (1990) · Zbl 0707.11029
[155] Lewis, D. W., The level of a Witt ring, Linear and Multilinear Algebra, 27, 163-165 (1990) · Zbl 0703.11020
[156] Lewis, D. W., (Levels of fields and quaternion algebras—a survey. Théorie des nombres, Années 1996/97-1997/98 (1999), Publ. Math. UFR Sci. Tech. Besançon, Univ. Franche-Comté: Publ. Math. UFR Sci. Tech. Besançon, Univ. Franche-Comté Besançon), 9
[157] Lewis, D. W.; Tignol, J.-P., Square class groups and Witt rings of central simple algebras, J. Algebra, 154, 360-376 (1993) · Zbl 0766.11024
[158] Lewis, D. W.; Tignol, J.-P., Classification theorems for central simple algebras with involution, Manuscripta Math., 100, 259-276 (1999) · Zbl 0953.11011
[159] Lorenz, F.; Leicht, J., Die Primideale des Wittschen ringes, Invent. Math., 10, 82-88 (1970) · Zbl 0227.13015
[160] Mammone, P.; Tignol, J.-P.; Wadsworth, A. R., Fields of characteristic 2 with prescribed \(u\)-invariants, Math. Ann., 290, 109-128 (1991) · Zbl 0713.12002
[161] Marshall, M. A., Classification of finite spaces of orderings, Can. J Math., 31, 320-330 (1979) · Zbl 0412.10012
[162] Marshall, M. A., Queen’s Papers in Pure and Applied Mathematics, (Abstract Witt Rings, vol. 57 (1980), Queen’s University: Queen’s University Kingston, Ontario)
[163] Marshall, M. A., Queen’s Papers in Pure and Applied Mathematics, (Bilinear Forms and Orderings on Commutative Rings, vol. 71 (1985), Queen’s University: Queen’s University Kingston, Ontario)
[164] Marshall, M. A., Exponentials and logarithms on Witt rings, Pacific J. Math., 127, 127-140 (1987) · Zbl 0638.10012
[165] Marshall, M. A., The elementary type conjecture in quadratic form theory, Contemp. Math., 344, 275-293 (2004) · Zbl 1143.11315
[166] Marshall, M. A.; Powers, V., Higher level form schemes, Commun. Algebra, 21, 4083-4102 (1993) · Zbl 0794.11020
[167] Marshall, M.; Yucas, J., Linked quaternionic mappings and their associated Witt rings, Pacific J. Math., 95, 411-425 (1981) · Zbl 0423.10013
[168] Merkurjev, A. S., On the norm residue symbol of degree 2, Dokl. Akad. Nauk SSSR, 261, 542-547 (1981), English Translation: Soviet Math. Dokl. 24 (1981), 546-551.
[169] Merkurjev, A. S., Kaplansky’s conjecture in the theory of quadratic forms, (Russian). Zap. Nauchn Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 175 (1989), Koltsai Moduli 3,75-89, 163-164; translation in J. Soviet Math. 57 (1991), no. 6, 3489-3497. · Zbl 0745.11026
[170] Merkurjev, A. S., Simple algebras and quadratic forms, (Russian). Izv. Akad. Nauk SSSR. Ser. Mat, 55, 218-224 (1991) · Zbl 0733.12008
[171] Merkurjev, A. S., (On the norm residue homomorphism of degree two, Proceedings of the St. Petersburg Mathematical Society, vol. XII (2006), American Mathematica Society: American Mathematica Society Providence, RI), 103-124, Am. Math. Soc. Transl. Ser. 2, 219 · Zbl 1180.19001
[172] Mestre, J.-F., Extensions régulières de \(Q (t)\) de groupe de Galois \(Ã_n\), J. Algebra, 131, 483-495 (1990) · Zbl 0714.11074
[173] Milnor, J., Algebraic K-Theory and Quadratic Forms, Invent. Math., 9, 318-344 (1970) · Zbl 0199.55501
[174] Milnor, J.; Husemoller, D., (Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Gren- zgebiete, vol. 73 (1973), Springer-Verlag: Springer-Verlag Berlin Heidelberg New York)
[175] Minác, J.; Spira, M., Witt rings and Galois groups, Ann. of Math., 144, 35-60 (1996) · Zbl 0861.11030
[176] Mináč, J.; Smith, T., Decomposition of Witt rings and Galois groups, Can. J Math., 47, 1274-1289 (1995) · Zbl 0853.11030
[177] Mináč, J.; Wadsworth, A. R., The “-invariant for algebraic extensions, Proc. Symp. Pure Math., 58, Part II, 333-358 (1995) · Zbl 0824.11018
[178] Minkowski, H., Ueber die Bedingungen, unter welchen zwei quadratische Formen mit rationalen Coeffizienten rational in einander transformirt werden knnen. (Auszug aus einem von Herrn H Minkowski in Bonn an Herrn Adolf Hurwitz gerichteten Briefe), J. Reine Angew. Math., 106, 5-26 (1890)
[179] Morel, F., Voevodsky’s proof of Milnor’s conjecture, Bull. Am. Math. Soc., 35, 123-143 (1998) · Zbl 0916.19002
[180] Ojanguren, M.; Parimala, R.; Sridharan, R., Symplectic bundles over affine surfaces, Comment. Math Helv., 61, 491-500 (1986) · Zbl 0598.14006
[181] O’Meara, O. T., Grundlehren der mathematischen Wissenschaften, (Introduction to Quadratic Forms, vol. 117 (2000), Springer-Verlag: Springer-Verlag Berlin Göttingen Heidelberg)
[182] Orlov, D.; Vishik, A.; Voevodsky, V., An exact sequence for \(K_*^M /2\) with applications to quadratic forms, Ann. of Math., 165, 1-13 (2007) · Zbl 1124.14017
[183] Osiak, K., A Cantor cube as a space of higher level orderings, Tatra Mount. Math. Publ., 32, 71-84 (2005) · Zbl 1150.11418
[184] Osiak, K., The Boolean space of higher level orderings, Fund. Math., 196, 101-117 (2007) · Zbl 1126.12002
[185] Osiak, K.; Sładek, A., Anote on number of orderings of higher level, Arch. Math., 86, 101-110 (2006) · Zbl 1084.11013
[186] Parimala, R., Witt groups of conics, elliptic, and hyperelliptic curves, J. Number Theory, 28, 69-93 (1988) · Zbl 0696.14016
[187] Parimala, R.; Sujatha, R., Witt groups of hyperelliptic curves, Comment. Math. Helv., 65, 559-580 (1990) · Zbl 0741.11023
[188] Parimala, R.; Sridharan, R., Non-surjectivity of the Clifford invariant map, Proc. Indian Acad. Sci (Math. Sci.), 104, 49-56 (1994) · Zbl 0808.11027
[189] Perlis, R.; Szymiczek, K.; Conner, P. E.; Litherland, R., Matching Witts with global fields, Contemp Math., 155, 365-387 (1994) · Zbl 0807.11024
[190] Pfister, A., Zur Darstellung von —1 als Summe von Quadraten in einem Körper, J. London Math Soc., 40, 159-165 (1965) · Zbl 0131.25002
[191] Pfister, A., Quadratische formen in beliebigen Körpern, Invent. Math., 1, 116-132 (1966) · Zbl 0142.27203
[192] Pfister, A., Zur Darstellung definiter Funktionen als summe von quadraten, Invent. Math., 4, 229-237 (1967) · Zbl 0222.10022
[193] Pfister, A., Neuere entwicklungen in der theorie der quadratischen formen, Jber. Deutsch. Math.-Verein., 74, 131-142 (1972) · Zbl 0245.10010
[194] Pfister, A., Über einige neuere ergebnisse aus der algebraischen theorie der quadratischen formen, Abh. Braunschw. Wiss. Gesellschaft, 32, 61-68 (1982) · Zbl 0502.10009
[195] A. Pfister, Some remarks on the historical development of the algebraic theory of quadratic forms CMS Conference Proceedings, vol. 4, American Mathematica Society, Providence, RI, 1984 pp. 1-16; Quadratic and Hermitian forms (Hamilton, Ontario, 1983).; A. Pfister, Some remarks on the historical development of the algebraic theory of quadratic forms CMS Conference Proceedings, vol. 4, American Mathematica Society, Providence, RI, 1984 pp. 1-16; Quadratic and Hermitian forms (Hamilton, Ontario, 1983).
[196] Pfister, A., Quadratische formen, Dok. Gesch. Math., 6, 657-671 (1990) · Zbl 0807.11002
[197] Pfister, A., Quadratic lattices in function fields of genus 0, Proc. London Math. Soc., 66, 3, 257-278 (1993) · Zbl 0808.11025
[198] Pfister, A., Lecture Notes Series, vol. 217: Quadratic Forms withApplications to Algebraic Geometry and Topology (1995), London Math. Soc. Cambridge Univ. Press: London Math. Soc. Cambridge Univ. Press Cambridge
[199] Pfister, A., on the milnor conjectures: history, influence, applications, Jber. Dt. Math.-Verein., 102, 15-41 (2000) · Zbl 1140.19300
[200] Pourchet, Y., Sur la représentation en somme de carrés des polynômes à une indéterminée sur un corps de nombres algébriques, Acta Arith., 19, 89-104 (1971) · Zbl 0244.10019
[201] Prestel, A., Remarks on the Pythagoras and Hasse number of real fields, J. Reine Angew. Math, 303/304, 284-294 (1978) · Zbl 0396.10013
[202] Prestel, A., On trace forms of algebraic function fields, Rocky Mountain J. Math., 19, 897-911 (1989) · Zbl 0702.11021
[203] Revoy, P., Niveaux d’ordre supérieur des corps et des anneaux, C.R. Acad. Sci. Paris Sér. I Math, 307, 203-206 (1988) · Zbl 0663.10013
[204] Scharlau, W., Grundlehren der mathematischen Wissenschaften, (Quadratic and Hermitian Forms, vol. 270 (1985), Springer-Verlag: Springer-Verlag Berlin Heidelberg New York Tokyo)
[205] Scharlau, W., On trace forms of algebraic number fields, Math. Zeit., 196, 125-127 (1987) · Zbl 0658.10025
[206] Scharlau, W., Generating functions of finitely generated Witt rings, Acta Arith., 54, 51-59 (1989) · Zbl 0693.10018
[207] Scharlau, W., On the history of the algebraic theory of quadratic forms, Contemp. Math., 272, 229-259 (2000) · Zbl 0974.11001
[208] Serre, J.-P., L’invariant de Witt de la forme tr \((x^2)\), Comment. Math. Helvetici, 59, 651-676 (1984) · Zbl 0565.12014
[209] Shapiro, D. B., Compositions of Quadratic Forms, (De Gruyter Expositions in Mathematics, vol. 32 (2000), W. de Gruyter & Co.: W. de Gruyter & Co. Berlin) · Zbl 0954.11011
[210] Shapiro, D. B.; Tignol, J.-P.; Wadsworth, A. R., Witt rings and Brauer groups under multiquadratic extensions, II. J. Algebra, 78, 58-90 (1982) · Zbl 0492.10015
[211] Sivatski, A. S., Nonexcellence of multiquadratic field extensions, J. Algebra, 275, 859-866 (2004) · Zbl 1133.11027
[212] Sivatski, A. S., Nonexcellence ofthe function field ofthe product of two conics, K-Theory, 34, 209-218 (2005) · Zbl 1123.11014
[213] Sładek, A., Witt rings of quaternion algebras, J. Algebra, 103, 267-272 (1986) · Zbl 0593.10017
[214] Sładek, A., Witt rings of complete skew fields. II, Wiss. Beitr. Martin-Luther-Univ. Halle-Wittenberg, Halle (Saale), 33, 239-243 (1986)
[215] Sładek, A., Witt rings of complete skew fields, Pacific J. Math., 132, 391-399 (1988) · Zbl 0607.10015
[216] Sładek, A., Witt rings of division algebras over global fields, Commun. Algebra, 18, 2159-2175 (1990) · Zbl 0725.11021
[217] Sładek, A., Witt rings of even degree division algebras over number fields, Commun. Algebra, 22, 3531-3543 (1994) · Zbl 0808.11026
[218] Sładek, A., Higher degree Harrison equivalence and Milnor K-functor, Acta Math. Inf. Univ. Ostrav, 6, 183-189 (1998) · Zbl 1024.11068
[219] Springer, T. A., Sur les formes quadratiques d’indice zéro, C.R. Acad. Sci. Paris, 234, 1517-1519 (1952) · Zbl 0046.24303
[220] Springer, T. A., Quadratic forms over a field with a discrete valuation, Indag. Math., 17, 352-362 (1955) · Zbl 0067.27605
[221] Szczepanik, L., Fields and quadratic form schemes with the index of radical not exceeding 16, Ann Math. Siles., 1, 23-46 (1985)
[222] Soviet Math. Doklady, 40, 355-358 (1990) · Zbl 0699.10032
[223] Szyjewski, M., On the Witt ring of a relative projective line, Coll. Math., 75, 53-78 (1998) · Zbl 0956.11011
[224] Szymiczek, K., Quadratic forms over fields with finite square class number, Acta Arith., 28, 195-221 (1975) · Zbl 0317.10034
[225] Szymiczek, K., Generalized Hilbert fields, J. Reine Angew. Math., 329, 58-65 (1981) · Zbl 0461.12013
[226] Szymiczek, K., Matching Witts locally and globally, Math. Slovaca, 41, 315-330 (1991) · Zbl 0766.11023
[227] Szymiczek, K., Witt equivalence of global fields, Commun. Algebra, 19, 1125-1149 (1991) · Zbl 0724.11020
[228] Szymiczek, K., Algebra, Logic and Applications, (Bilinear Algebra. An introduction to the algebraic theory of quadratic forms, vol. 7 (1997), Gordon and Breach Science Publishers: Gordon and Breach Science Publishers Amsterdam) · Zbl 0127.01901
[229] Szymiczek, K., Hilbert-symbol equivalence of number fields, Tatra Mount. Math. Publ., 11, 7-16 (1997) · Zbl 0978.11012
[230] Szymiczek, K., Acharacterizationoftame Hilbert-symbol equivalence, Acta Math. Inf. Univ. Ostrav, 6, 191-201 (1998) · Zbl 1024.11022
[231] Szymiczek, K., Tame equivalence and wild sets, Semigroup Forum, 60, 260-270 (2000) · Zbl 0946.11028
[232] Szymiczek, K., Ten problems on quadratic forms, Acta Math. Inf. Univ. Ostrav., 10, 133-143 (2002) · Zbl 1080.11033
[233] Tsen, C. C., Zur Stufentheorie der quasialgebraisch-Abgeschlossenheit kommutativer Körper, J. Chin. Math. Soc., 1, 81-92 (1936) · Zbl 0015.38803
[234] Vishik, A., On dimensions of quadratic forms, Dokl. Akad. Nauk, 373, 4, 445-447 (2000) · Zbl 1039.11018
[235] Vishik, A., Fields of u-invariant 2\(^r\) + 1. Preprint No. 229, October 2006. Linear Algebraic Groups and Related Structures preprint server
[236] Voevodsky, V., On 2-torsion in motovic cohomology. Preprint 2001
[237] Wadsworth, A. R., Similarity of quadratic forms and isomorphism of their function fields, Trans Am. Math. Soc., 208, 352-358 (1975) · Zbl 0336.15013
[238] Wadsworth, A. R., Merkurjev’s elementary proof of Merkurjev’s theorem, Contemp. Math., 55, 741-776 (1986) · Zbl 0604.16022
[239] Waterhouse, W. C., Scaled trace forms over number fields, Arch. Math., 47, 229-231 (1986) · Zbl 0607.10013
[240] Witt, E., Theorie der quadratischen Formen in beliebigen Körpern, J. Reine Angew. Math., 176, 31-44 (1937) · JFM 62.0106.02
[241] Yucas, J. L., Quadratic forms and radicals of fields, Acta Arith., 39, 313-322 (1981) · Zbl 0471.10015
[242] Craven, T. C., Orderings, valuations, and hermitian forms over *-fields, Proc. Symp. Pure Math., 58, Part II, 149-160 (1995) · Zbl 0822.11029
[243] Elman, R.; Prestel, A., Reduced stability of the Witt ring of a field and its Pythagorean closure, Am J. Math., 106, 1237-1260 (1984) · Zbl 0573.10015
[244] Fitzgerald, R. W.; Yucas, J., Combinatorial techniques and abstract Witt rings, I. J. Algebra, 114, 40-52 (1988) · Zbl 0641.10018
[245] Fitzgerald, R. W.; Yucas, J., Combinatorial techniques and abstract Witt rings, II, Rocky Mountain J Math., 19, 687-708 (1989) · Zbl 0705.11021
[246] Jannsen, U.; Sujatha, R., Levels of function fields of surfaces over number fields, J. Algebra, 251, 1, 350-357 (2002) · Zbl 1069.11025
[247] Kaplansky, I.; Shaker, J., Abstract quadratic forms, Can. J. Math., 21, 1218-1233 (1969) · Zbl 0238.15010
[248] Knus, M.-A.; Lam, T. Y.; Shapiro, D. B.; Tignol, J.-P., Discriminants of involutions on biquaternion algebras, Proc. Symp. Pure Math., 58, Part II, 279-303 (1995) · Zbl 0828.11023
[249] Lam, T. Y., The Algebraic Theory of Quadratic Forms (1973), W.A. Benjamin, Addison-Wesley: W.A. Benjamin, Addison-Wesley Reading, Mass., Second printing with revisions, 1980. · Zbl 0259.10019
[250] Lam, T. Y.; Leep, D. B.; Tignol, J.-P., Biquaternion algebras and quartic extensions, Math. Ann., 62, 272-285 (1991)
[251] Lorenz, F., Lecture Notes in Mathematics, vol. 130: Quadratische Formen über Körpern (1970), Springer-Verlag: Springer-Verlag Berlin-Heidelberg-New York, 2007 · Zbl 0211.35303
[252] Mammone, P.; Shapiro, D. B., The Albert quadratic form for an algebra of degree four, Proc. Am. Math. Soc., 105, 525-530 (1989) · Zbl 0669.10042
[253] Prestel, A., Lecture Notes in Mathematics, (Lectures on Formally Real Fields, vol. 1093 (1984), Springer-Verlag: Springer-Verlag Berlin Heidelberg New York Tokyo)
[254] Shapiro, D. B., Products of sums of squares, Expos. Math., 2, 235-261 (1984) · Zbl 0541.10025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.