×

zbMATH — the first resource for mathematics

On trace forms of algebraic number fields. (English) Zbl 0658.10025
Let L/K be a finite separable field extension and let \(Tr_{L/K}<1>\) denote the associated trace form on \(L\times L\) over K given by \((x,y)\mapsto Tr_{L/K}(xy)\). For any ordering P of K the signature \(sign_ P(Tr_{L/K}<1>)\) is nonnegative. In the paper under review it is shown that for algebraic number fields K the converse is true: Theorem: Let K be an algebraic number field with Witt ring W(K). For any \(\phi\in W(K)\) assume \(sign_ P(\phi)\geq 0\) for all orderings P of K. Then there exists a finite extension L/K such that \(\phi\) is Witt equivalent to \(Tr_{L/K}<1>.\)
This generalizes similar results obtained by Conner and Perlis in the case \(K={\mathbb{Q}}\). The proof uses a certain version of Hilbert’s irreducibility theorem.
Reviewer: H.-J.Bartels

MSC:
11E16 General binary quadratic forms
12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
11E12 Quadratic forms over global rings and fields
PDF BibTeX Cite
Full Text: DOI EuDML
References:
[1] Conner, P.E., Perlis R.: A survey of trace forms of algebraic number fields. World Scientific, Singapore 1984 · Zbl 0551.10017
[2] Dries. L.v.d.: Model theory of fields. Thesis, Utrecht 1978
[3] Estes, D.R., Hurrelbrink, J., Perlis, R.: Total positivity and algebraic Witt classes. Comment. Math. Helvetici60, 284-290 (1985) · Zbl 0589.10021
[4] Ischebeck, F., Scharlau, W.: Hermitesche and orthogonale Operatoren über kommutativen Ringen. Math. Ann.200, 327-334 (1973) · Zbl 0265.18010
[5] Krüskemper, M.: Diplomarbeit, Münster 1986
[6] Scharlau, W.: Unzerlegbare quadratische Formen. Unpublished preprint 1981
[7] Scharlau, W.: Quadratic and Hermitian Forms. Berlin Heidelberg New York: Springer 1985 · Zbl 0584.10010
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.