On trace forms of étale algebras and field extensions
The investigation of quadratic forms represented as trace forms of finite extension fields was initiated by \textit{P. E. Conner} and \textit{R. Perlis} [A survey of trace forms of algebraic number fields. World Sci., Singapore (1984; Zbl 0551.10017)]. The emphasis was on the characterization of forms which are similar (Witt-equivalent) to trace forms of field extensions and étale algebras. This line of investigation has been refined by seeking the characterization of forms which are isometric rather than similar to trace forms of field extensions and étale algebras. It turned out that most of the results characterizing quadratic forms up to similarity hold also up to isometry of quadratic forms. The present authors confirm this rule by strengthening an earlier result of A. Prestel. They prove that for a Hilbertian field \(K\) of characteristic zero the trace form of an étale algebra over \(K\) of dimension \(>2\) is isometric to the trace form of some field extension of \(K\). The authors also include several observations on trace forms of étale algebras. Thus they determine forms of this type of dimensions \(<6\) in terms of quadratic form invariants and they relate the existence of trace forms to the field invariants. An open problem asks if every trace form of an étale algebra can be realized as the direct orthogonal sum of trace forms of two-power degree field extensions.
scientific article; zbMATH DE number 737171
scientific article