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Quadratic forms over quadratic extensions of generalized local fields. (English) Zbl 0652.10016

In this paper the behaviour under quadratic extensions of special assumptions about Pfister forms is studied. Thus a field F is called n- local, \(n\geq 2\), if there are exactly 2 isometry classes of n-fold Pfister forms over F. K. Szymiczek [J. Reine Angew. Math. 329, 58- 65 (1981; Zbl 0461.12013)] has shown that F is n-local if and only if it is (n-1)-Hilbert: For any (n-1)-fold Pfister form \(\Phi\) we have \([\dot F : D_ F(\Phi)]=1\) or 2, with 2 occuring. The author deals with the question of classification of n-fold Pfister forms over quadratic extensions of n-local fields; the case \(n=2\) having been dealt with by C. M. Cordes and J. Ramsey [Can. J. Math. 31, 1047-1058 (1979; Zbl 0426.10021)] by different methods.
In order to state the results we need some notation. For a field F of characteristic not 2, let \(p_ n(F)\) be the number of isometry classes of n-fold Pfister forms over F. The nth radical of F, written \(R_ nF\), is the intersection of value groups of all n-fold Pfister forms over F.
The main results of this paper are contained in following theorems: Theorem 1. Let F be an n-local field, i.e., \(p_ n(F)=2\) and let \(K=F(a^{1/2})\), where \(a\in F^*\setminus F^{*2}\). (i) If \(a\in R_{n-1}F\), then \(p_ n(K)=4\). (ii) If \(a\not\in R_{n-1}F\), and F is non-real, then \(p_ n(K)=2\). (iii) If \(a\not\in R_{n-1}F\), and F is formally real, then \(p_ n(K)=1.\)
Theorem 2. Let F be a non-real n-Hilbert field, H be a subgroup of \(F^*\) and \(K=F(\sqrt{H})\). Then K is an n-Hilbert field if and only if \((H\cap R_ nF)\subset F^{*^ 2}\) and \([H:(H\cap F^{*^ 2})]<\infty.\)
Theorem 3. Let F be a formally real n-Hilbert field, H be a subgroup of \(F^*\), and \(K=F(\sqrt{H})\). Then K is n-Pythagorean, i.e., every n-fold Pfister form represents all sums of squares, and if H is not contained in \(F^{*^ 2}\), then K is not n-Hilbert.
In the special case, when the ground field F is non-real and its nth radical is trivial, the following generalization of Theorem 2 is valid. Theorem 4. Let F be a non-real n-Hilbert field, \(R_ nF^{*^ 2}\), and K be a 2-extension of F. Then K is n-Hilbert if and only if the index \([K:F]\) is finite.
Reviewer: A.F.T.W.Rosenberg

MSC:

11E04 Quadratic forms over general fields
11E16 General binary quadratic forms
12F05 Algebraic field extensions
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References:

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