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The Witt invariant of the form \(\text{Tr}(x^ 2)\). (L’invariant de Witt de la forme \(\text{Tr}(x^ 2)\).) (French) Zbl 0565.12014
Let \(K\) be a commutative field of characteristic \(\neq 2\). Let \(\Gamma_ K\) be the Galois group of a separable closure of \(K\). Let \(E\) be an étale commutative \(K\)-algebra of rank \(n\) and \(d_ E\) its discriminant. Let \(w_ 2(Q_ E)\) be the Witt invariant of the quadratic form \(Q_ E(x)=Tr_{E/K}(x^ 2)\). Let \(e: \Gamma_ K\to S_ n\) be the homomorphism defined by \(E\), up to conjugation, where \(S_ n\) is the permutation group of n letters. Let \(s_ n\in H^ 2(S_ n,{\mathbb{Z}}/2{\mathbb{Z}})\) be the second Stiefel-Whitney class of the vector bundle over the classifying space of \(S_ n\) associated to the representation of \(S_ n\) in the orthogonal group \(O_ n({\mathbb{R}})\). The main theorem of this paper asserts that \(w_ 2(Q_ E)=e^*(s_ n)+(2,d_ E)\) as elements of \(\text{Br}_ 2(K)\simeq H^ 2(\Gamma_ K,{\mathbb{Z}}/2{\mathbb{Z}}).\)
Let \(\tilde S_ n\) be the extension of \(S_ n\) with kernel \({\mathbb{Z}}/2{\mathbb{Z}}\) defined by \(s_ n\). Since \(e^*(s_ n)\) can be viewed as the obstruction to an embedding problem defined from the exact sequence \(1\to {\mathbb{Z}}/2{\mathbb{Z}}\to \tilde S_ n\to S_ n\to 1\), the theorem shows how the computation of this obstruction is related to the computation of the Witt invariant of the quadratic form \(Q_ E\). Some examples and applications are given. In particular, extensions of degree 4 or 5 and extensions defined by polynomials of the type \(X^ n+aX+b\) are considered. Finally, the main theorem is generalized to quadratic forms \(Tr_{E/K}(\alpha X^ 2)\), where \(\alpha\) is a unit of \(E\).
Reviewer: N.Vila

12G05 Galois cohomology
11E04 Quadratic forms over general fields
11E81 Algebraic theory of quadratic forms; Witt groups and rings
57R20 Characteristic classes and numbers in differential topology
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