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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

##### MSC:
 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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