Geodesic
Spaces of Orderings IV
Canadian journal of mathematics, Tome 32 (1980) no. 3, pp. 603-627

Voir la notice de l'article provenant de la source Cambridge University Press

A major goal of this paper is to give a proof of the following isotropy criterion: Let X = (X,G) be a space of orderings in the terminology of [9] or [10], and let f be a form defined over G.Then f is anisotropic over X if and only if f is anisotropic over some finite subspace of X.This is the content of Theorem 1.4, and generalizes [1, Corollary 3.4]. Moreover, in view of the known structure of finite spaces (see [9]), this has, essentially, the strength of [2, Satz 3.9] or [12, Theorem 8.12]. The technique used to prove this criterion is roughly patterned on that of [6], and yields some interesting by-products: An interesting invariant of a space of orderings called the chain length is introduced (Definition 1.1) and spaces of orderings with finite chain length are classified (Theorem 1.6). 
Marshall, Murray. Spaces of Orderings IV. Canadian journal of mathematics, Tome 32 (1980) no. 3, pp. 603-627. doi: 10.4153/CJM-1980-047-0
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[1] 1. Becker, E. and Brôcker, L., On the description of the reduced Witt ring, J. Alg. 52 (1978), 328–346. Google Scholar

[2] 2. Brôcker, L., Zur théorie der quadratischen formen ilber formal reellen Kôrpern, Math. Ann. 210 (1974), 223, 256. Google Scholar

[3] 3. Brôcker, L., Characterization of fans and hereditarily pythagorian fields, Math. Z. 152 (1976), 149–163. Google Scholar

[4] 4. Brôcker, L., tJber die anzahl der anordnungen eines kommutativen kôrpers, Archiv der Math. 29 (1977), 458–464. Google Scholar

[5] 5. Brown, R., The reduced Witt ring of a formally real field, Trans. Amer. Math. Soc. 230 (1977), 257–292. Google Scholar

[6] 6. Brown, R. and Marshall, M., The reduced theory of quadratic forms, to appear, Rocky Mtn. J. of Math. Google Scholar

[7] 7. Craven, T., Characterizing reduced Witt rings of fields, J. of Alg. 58 (1978), 68–77. Google Scholar

[8] 8. Kleinstein, J. and Rosenberg, A., Succinct and representational Witt rings, to appear, Can. J. Math. Google Scholar

[9] 9. Marshall, M., Classification of finite space of orderings, Can. J. Math. 31 (1979), 320–330. Google Scholar

[10] 10. Marshall, M., Quotients and inverse limits of spaces of orderings, Can. J. Math. 31 (1979), 604–616. Google Scholar

[11] 11. Marshall, M., The Witt ring of a space of orderings, Trans. Amer. Math. Soc. 258 (1980), 505–521. Google Scholar

[12] 12. Prestel, A., Lectures on formally real fields, Lecture Notes, IMPA, Rio de Janeiro (1976). Google Scholar

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