Formally real fields with prescribed invariants in the theory of quadratic forms. (English) Zbl 0734.11028
In 1989 A. Merkur’ev solved an old problem on the u-invariant constructing non-real fields F such that \(I^ 3(F)=0\) and \(u(F)=2n\), where n is any given integer \(\geq 1\). The author extends the ideas of Merkur’ev to provide the readers with examples of formally real fields \(F_ 1\) and \(F_ 2\) such that \(I_ t^ 3(F_ 1)=I_ t^ 3(F_ 2)=0\), the Hasse numbers \(\tilde u(F_ 1)\) and \(\tilde u(F_ 2)\) are equal to 2n and \(u(F_ 1)=u(F_ 2)+2=2n\). In the construction she also takes care of other invariants connected with orderings, for example \(\tilde ud(F)\) which is a counterpart of \(\tilde u(F)\) in the family of quadratic forms \(q\cong q_+\perp -q_-\) with \(q_+,q_-\) totally positive.
Reviewer: A.Sładek (Katowice)
MSC:
| 11E04 | Quadratic forms over general fields |
| 11E10 | Forms over real fields |
| 12D15 | Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) |
Keywords:
division algebra; Clifford algebra; u-invariant; examples of formally real fields; Hasse numbers; quadratic formsReferences:
| [1] | Baeza, R.; Leep, D.; O’Ryan, M.; Prieto, J. P., Sums of squares of linear forms, Math. Z., 193, 297-306 (1986) · Zbl 0598.10032 |
| [2] | Elman, R.; Lam, T. Y., Classification Theorems for quadratic forms over fields, Comm. Math. Helv., 49, 373-381 (1974) · Zbl 0297.15024 |
| [3] | Elman, R., Quadratic forms and the \(u\)-invariant, III, Conference on Quadratic forms, Queen’s Papers in Pure and Applied Mathematics, 46, 422-444 (1977) · Zbl 0387.10012 |
| [4] | Elman, R.; Lam, T. Y.; Prestel, A., On some Hasse principles over formally real fields, Math. Z., 134, 291-301 (1973) · Zbl 0277.15013 |
| [5] | Hornix, E. A.M., Totally indefinite quadratic forms over formally real fields, Indagationes Math., 47, 305-312 (1985) · Zbl 0595.10016 |
| [6] | Hornix, E. A.M., Invariants of formally real fields and quadratic forms, Indagationes Math., 48, 47-60 (1986) · Zbl 0601.10015 |
| [7] | Hornix, E. A.M., Fed-linked fields and formally real fields satisfying \(B_3\), Indagationes Math., 1, 303-322 (1990), New Series · Zbl 0724.12003 |
| [8] | Hornix, E. A.M., Examples of fields with u˜ ≠ \(u\) (july 1989), University Utrecht, Preprint 572 |
| [9] | Lam, T. Y., The algebraic theory of quadratic forms (1973, 1980), W.A. Benjamin: W.A. Benjamin Reading Mass. · Zbl 0259.10019 |
| [10] | Lam, T. Y., Fields of \(u\)-invariant 6 after A. Merkurjev. Ring theory 1989, (Israel Mathematical Conference Proceedings (1989), The Weizmann Science Press: The Weizmann Science Press Israel), 12-30, in honor of S.A. Amitsur · Zbl 0683.10018 |
| [11] | Lam, T. Y., Manuscript, Some consequences of Merkurjev’s work on function fields (1989) |
| [12] | Mammone, P.; Tignol, J.-P., Clifford division algebras and anisotropic quadratic forms: two counterexamples, Glasgow Math. J., 28, 227-228 (1986) · Zbl 0597.16018 |
| [14] | Merkurjev, A., Simple algebras over function fields of quadrics, Manuscript (1989) |
| [15] | Scharlau, W., Quadratic and Hermitian forms (1985), Springer Verlag: Springer Verlag Berlin Heidelberg New York Tokyo · Zbl 0584.10010 |
| [16] | Tignol, J.-P., An elementary proof of Merkurjev’s theorem on function fields of quadrics, Manuscript (1989) |
| [17] | Ware, R., Hasse Principles and the \(u\)-invariant over formally real fields, Nagoya Math. J., 61, 117-125 (1976) · Zbl 0311.10028 |
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.
