The reduced Witt ring of a formally real field

Brown, Ron

doi:10.1090/S0002-9947-1977-0472781-4

The reduced Witt ring of a formally real field
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Trans. Amer. Math. Soc. 230 (1977), 257-292 Request permission

Abstract:

The reduced Witt rings of certain formally real fields are computed here in terms of some basic arithmetic invariants of the fields. For some fields, including the rational function field in one variable over the rational numbers and the rational function field in two variables over the real numbers, this is done by computing the image of the total signature map on the Witt ring. For a wider class of fields, including all those with only finitely many square classes, it is done by computing the Witt rings of certain ultracompletions of the field and representing the reduced Witt ring as an appropriate subdirect product of the Witt rings of the ultracompletions. The reduced Witt rings of a still wider class of fields, including for example the fields of transcendence degree one and the rational function field in three variables over the real numbers, are computed similarly, except that the description of the subdirect product no longer involves only local conditions.

References

Shreeram Abhyankar, Ramification theoretic methods in algebraic geometry, Annals of Mathematics Studies, No. 43, Princeton University Press, Princeton, N.J., 1959. MR 0105416
Ron Brown, Real places and ordered fields, Rocky Mountain J. Math. 1 (1971), no. 4, 633–636. MR 285512, DOI 10.1216/RMJ-1971-1-4-633
Ron Brown, An approximation theorem for extended prime spots, Canadian J. Math. 24 (1972), 167–184. MR 292802, DOI 10.4153/CJM-1972-015-3
Ron Brown, Valuations, primes and irreducibility in polynomial rings and rational function fields, Trans. Amer. Math. Soc. 174 (1972), 451–488. MR 371872, DOI 10.1090/S0002-9947-1972-0371872-1
Ron Brown, Extended prime spots and quadratic forms, Pacific J. Math. 51 (1974), 379–395. MR 392960
Ron Brown, Superpythagorean fields, J. Algebra 42 (1976), no. 2, 483–494. MR 427286, DOI 10.1016/0021-8693(76)90109-5
Thomas C. Craven, The topological space of orderings of a rational function field, Duke Math. J. 41 (1974), 339–347. MR 349639
Thomas C. Craven, The Boolean space of orderings of a field, Trans. Amer. Math. Soc. 209 (1975), 225–235. MR 379448, DOI 10.1090/S0002-9947-1975-0379448-X
Richard Elman and T. Y. Lam, Quadratic forms over formally real fields and pythagorean fields, Amer. J. Math. 94 (1972), 1155–1194. MR 314878, DOI 10.2307/2373568
Richard Elman, Tsit Yuen Lam, and Alexander Prestel, On some Hasse principles over formally real fields, Math. Z. 134 (1973), 291–301. MR 330045, DOI 10.1007/BF01214693
Irving Kaplansky, Maximal fields with valuations, Duke Math. J. 9 (1942), 303–321. MR 6161
Manfred Knebusch, Alex Rosenberg, and Roger Ware, Signatures on semilocal rings, J. Algebra 26 (1973), 208–250. MR 327761, DOI 10.1016/0021-8693(73)90021-5
John Milnor, Algebraic $K$-theory and quadratic forms, Invent. Math. 9 (1969/70), 318–344. MR 260844, DOI 10.1007/BF01425486
John Milnor and Dale Husemoller, Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73, Springer-Verlag, New York-Heidelberg, 1973. MR 0506372
Albrecht Pfister, Quadratische Formen in beliebigen Körpern, Invent. Math. 1 (1966), 116–132 (German). MR 200270, DOI 10.1007/BF01389724
A. Prestel, Quadratische Semi-Ordnungen und quadratische Formen, Math. Z. 133 (1973), 319–342 (German). MR 337913, DOI 10.1007/BF01177872
Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0152834
Paulo Ribenboim, Théorie des valuations, Séminaire de Mathématiques Supérieures, No. 9 (Été, vol. 1964, Les Presses de l’Université de Montréal, Montreal, Que., 1968 (French). Deuxième édition multigraphiée. MR 0249425
O. F. G. Schilling, The Theory of Valuations, Mathematical Surveys, No. 4, American Mathematical Society, New York, N. Y., 1950. MR 0043776
Ludwig Bröcker, Zur Theorie der quadratischen Formen über formal reellen Körpern, Math. Ann. 210 (1974), 233–256 (German). MR 354549, DOI 10.1007/BF01350587

Lectures on formally real fields