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Total positivity and algebraic Witt classes. (English) Zbl 0589.10021
Let E be a finite extension of the algebraic number field F. \(<E>\) denotes the class in the Witt ring W(F) of F given by the trace form \(tr_{E/F}(x^ 2)\). The Witt classes in W(F) arising in this way from algebraic extensions E/F are called algebraic classes.
The main result of the paper under review is the following theorem: The element a in \(F^{\#}\) is totally positive in F (i.e. a is positive in every possible ordering of F) if and only if the Witt class of the quadratic form \(a\cdot X^ 2\) in W(F) is algebraic.
The proof of the above mentioned theorem uses results of Conner and Perlis concerning trace forms of algebraic number fields and a characterization of totally positive elements in \(F^{\#}\) proved in the first section of the article.
Reviewer: H.-J.Bartels

11E12 Quadratic forms over global rings and fields
11R80 Totally real fields
11R70 \(K\)-theory of global fields
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