Invariant de Clifford équivariant d'un caractère orthogonal. (Clifford invariant equivariant to an orthogonal character) (Q1821139)

A \(\Gamma\)-invariant quadratic form (M,h) over K has a Clifford invariant in the equivariant Brauer group B(K,\(\Gamma)\), which splits as \(B(K)\times H^ 2(\Gamma;K^{\times})\). The image c(\(\theta)\) of the second component in \(H^ 2(\Gamma;\bar K^{\times})\cong H^ 2(\Gamma; {\mathbb{Q}}/ {\mathbb{Z}})\) depends only on the character \(\theta\) of M. It is shown that c(\(\theta)\) is the natural image of the Stiefel-Whitney class \(SW(\theta)\in H^ 2(\Gamma;\pm 1)\). Using representation theory methods, the calculation of c(\(\theta)\) is reduced (when the Sylow 2- subgroup of \(\Gamma\) is metacyclic) to the cases of dihedral and biquadratic (sums of two distinct nontrivial abelian) characters; in this case the main result asserts that \(c(\theta)=-1\) if and only if Ker \(\theta\cap [\Gamma,\Gamma]\) has odd order. This is applied to show that if K is a p-adic field, N a Galois extension with group \(\Gamma\) and \(\theta\) a dihedral or biquadratic character of \(\Gamma\), then \(c(\theta)=(-1)^ e\), where e is the ramification index of N over the fixed field of Ker \(\theta\).