Total positivity and algebraic Witt classes (Q1073826)

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.