Formally real fields with prescribed invariants in the theory of quadratic forms
In 1989 A. Merkur'ev solved an old problem on the u-invariant constructing non-real fields F such that \(I^ 3(F)=0\) and \(u(F)=2n\), where n is any given integer \(\geq 1\). The author extends the ideas of Merkur'ev to provide the readers with examples of formally real fields \(F_ 1\) and \(F_ 2\) such that \(I_ t^ 3(F_ 1)=I_ t^ 3(F_ 2)=0\), the Hasse numbers \(\tilde u(F_ 1)\) and \(\tilde u(F_ 2)\) are equal to 2n and \(u(F_ 1)=u(F_ 2)+2=2n\). In the construction she also takes care of other invariants connected with orderings, for example \(\tilde ud(F)\) which is a counterpart of \(\tilde u(F)\) in the family of quadratic forms \(q\cong q_+\perp -q_-\) with \(q_+,q_-\) totally positive.
scientific article; zbMATH DE number 4214117
scientific article