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On trace forms of algebraic number fields. (English) Zbl 0777.11009
Let \(K\) be a field of characteristic \(\neq 2\), \(A/K\) a finite dimensional commutative and étale algebra over \(K\). The quadratic form \(A\to K: x\mapsto \text{tr}_{A/K}(x^ 2)\) is called trace form. Let \(\psi\) be a nondegenerate quadratic form of dimension \(n\geq 4\) over \(K\). We prove the following. If \(K\) is an algebraic number field, then there exists a field extension \(L/K\) such that the trace form of \(L/K\) is isometric to \(\psi\) if and only if no signature of \(\psi\) is negative. Each trace form of an étale algebra of odd dimension \(\geq 3\) defined over a Hilbertian field \(K\) of characteristic 0 is a trace form of a field extension.

11E12 Quadratic forms over global rings and fields
11E04 Quadratic forms over general fields
11E81 Algebraic theory of quadratic forms; Witt groups and rings
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[1] P.Conner and R.Perlis, A Survey of Trace Forms of Algebraic Number Fields. Singapore 1984. · Zbl 0551.10017
[2] M. Epkenhans, Spurformen ?ber lokalen K?rpern. Schriftenreihe Math. Inst. Univ. M?nster, Ser. 244, (1987). · Zbl 0611.10012
[3] M. Epkenhans, Trace Forms of Dyadic Number Fields. J. Number Theory38, 359-365 (1991). · Zbl 0731.11021
[4] M.Epkenhans, On Trace Forms of Algebraic Number Fields. Preprint 1992. · Zbl 0777.11009
[5] M. Kr?skemper, Algebraic number field extensions with prescribed trace form. J. Number Theory40, 120-124 (1992). · Zbl 0762.11014
[6] J. F. Mestre, Extensions r?guli?res de 2e(t) de group de Galois ?tA n . J. Algebra131, 483-495 (1990). · Zbl 0714.11074
[7] A. Prestel, On trace forms of algebraic function fields. Rocky Mountain J. Math.19, 897-911 (1989). · Zbl 0702.11021
[8] W. Scharlau, On Trace Forms of Algebraic Number Fields. Math. Z.196, 125-127 (1987). · Zbl 0658.10025
[9] O. Taussky, The Discriminant Matrices of an Algebraic Number Field. J. London Math. Soc. (2)43, 152-154 (1968). · Zbl 0155.37903
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