Galois cohomology and the trace form. (English) Zbl 0804.12003

This is a very readable survey on recent progress concerning trace forms and Galois cohomology. Let \(K\) be a field of characteristic \(\neq 2\), let \(f\in K[X]\) be a separable polynomial and let \(L= K[X]/ (f)\). The trace form \(q_ L: L\to K\), defined by \(q_ L(x):= \text{Tr}_{L/K} (x^ 2)\), is a nondegenerate quadratic form over the \(K\)-vector space \(L\). The author first reviews results from early history of trace forms, cohomological invariants as quadratic form to recent results of Serre (unpublished) on cohomological invariants of an étale algebra, as well as applications of the vanishing formula for the Stiefel-Whitney classes, on solvability of certain Galois embedding problems. Results concerning which quadratic forms are trace forms are also considered. The last part is devoted to trace forms of Galois extensions, the aim being to describe the isomorphism class of \(q_ L\) as \(G\)-form, if \(L/K\) is a Galois extension with Galois group \(G\), in terms of cohomological invariants.
Reviewer: N.Vila (Barcelona)


12G05 Galois cohomology
11R34 Galois cohomology
11E04 Quadratic forms over general fields
12-02 Research exposition (monographs, survey articles) pertaining to field theory