Hermitian forms and the \(u\)-invariant
The \(u\)-invariant is one of the most interesting invariants in the algebraic theory of quadratic forms. This invariant was introduced by Kaplansky over non formally real fields and by Elman and Lam over formally real fields. (In this work it is always assumed that characteristic \(\not= 2\)). In this work, the author adapts this to the context of Hermitian forms over a division algebra with involution. Let \(D\) be a division algebra with an involution \(\sigma\) and let \(\varepsilon\) be an element of the center of \(D\) with \(\sigma(\varepsilon)\varepsilon = 1\). The \(u\)-invariant with respect to \(\varepsilon\), is by definition the supremum over the dimension of anisotropic \(\varepsilon\)-Hermitian forms over \((D, \sigma)\). This number is denoted by \(u(D, \sigma, \varepsilon)\) (value \(\infty\) is admitted also). The author studies the notion of Hermitian \(u\)-invariant. He gives some estimates of the \(u\)-invariant of a division algebra with involution in terms of the \(u\)-invariant of some subalgebras stable under the involution. He also finds some finiteness results for comparing the \(u\)-invariant of a division algebra with involution and that of its center. Some results about the values of this invariant are also given. A description of the Tits index of some algebraic groups of classical type over \(\mathbb Q_p(t)\), \(p \neq 2\) is given as an application.
scientific article; zbMATH DE number 2168084
scientific article