On the cohomology of biquadratic extensions
Let \(F\) be a field of characteristic \(\neq 2\) and \(E/F\) a biquadratic Galois extension. Relationships between the Galois cohomology (with coefficients \(\mathbb{Z}/ 2\mathbb{Z}\)) of \(F\), \(E\) and the three intermediate quadratic extensions have been studied by various authors. In particular, Merkurjev and Tignol have introduced two series of seven-term complexes \(S_ n\) and \(S^ n\) \((n\geq 0)\) involving these cohomology groups, but too technical to be written down here, and have proved their exactness in some cases, e.g. for \(n\leq 1\). The main result of this paper is that \(S_ n\) and \(S^ n\) are exact sequences in the following cases: (1) \(n=2\). (2) The Milnor conjecture holds for \(F\) and all its finite extensions in degree \(\leq n\), and a) \(F= F^ 2+ F^ 2\) or b) \(E\) is Pythagorean. Recall that \(K\) is Pythagorean if the set of squares of \(K\) is closed under addition, and that the Milnor conjecture in degree \(n\) for \(K\) predicts that the natural homomorphism \(K_ n^ M(K)/2\to H^ n(K, \mathbb{Z}/2)\) is an isomorphism.
scientific article; zbMATH DE number 590900