Galois cohomology of biquadratic extensions
Let \(F\) be a field of characteristic different from 2, and let \(\Gamma\) be the Galois group of the separable algebraic closure \(\overline{F}\) over \(F\). Let \(\Lambda\) be the field of two elements, considered as a \(\Gamma\)-module by the trivial action of \(\Gamma\); and if \(K\) is a subfield of \(\overline{F}\) which is a normal, finite dimensional extension of \(F\), let \(H^ n K\) denote the cohomology group \(H^ n(\Gamma',\Lambda)\) for the corresponding subgroup \(\Gamma'\) of \(\Gamma\). Note that \(H^ n K\) is isomorphic to \(H^ n(\Gamma, \Lambda[\Gamma/\Gamma'])\) for the group algebra of the Galois group \(\Gamma/\Gamma'\) of \(K\) over \(F\). Now let \(M=F[\sqrt{a_ 1}, \sqrt{a_ 2}, \sqrt{a_ 3}]\) be a biquadratic extension of \(F\) for elements \(a_ 1\), \(a_ 2\), \(a_ 3\) of \(F\) such that \(a_ 1 a_ 2 a_ 3=1\), and assume that \(\sqrt{a_ 1} \sqrt{a_ 2} \sqrt{a_ 3}=1\). Let \(G\) be the Galois group of \(M\) over \(F\), and for \(i=1,2,3\), let \(L_ i=F[\sqrt{a_ i}]\) and let \(G_ i\) be the Galois group of \(L_ i\) over \(F\). The authors construct a long exact sequence of \(\Gamma\)-modules: \[ 0 \longrightarrow \Lambda \overset \text{res} \longrightarrow \Lambda[G] \longrightarrow \bigoplus ^ 3_{i=1} \Lambda[G_ i] \longrightarrow \Lambda[G] \overset \text{cor} \longrightarrow \Lambda, \] where res maps 1 to the sum of elements of \(G\) and induces the restriction map on cohomology groups, and cor is the augmentation map of the group algebra and induces the corestriction map on cohomology groups. By decomposing this exact sequence and a dual exact sequence of \(\Gamma\)- modules into short exact sequences and taking maps and composites of maps from the resulting long exact sequences of cohomology, the authors obtain the following complexes: \[ \begin{multlined} S_ n: H^ n M\oplus (H^ n F)^ 3 \to \bigoplus^ 3_{i=1} H^ n L_ i\to (H^ n F)^ 2\to H^{n+1} F\overset\text{res}\longrightarrow\\ \overset \text{res} \longrightarrow H^{n+1} {M} \bigoplus^ 3_{i=1} H^{n+1} L_ {i} H^{n+1} M\oplus (H^{n+1} F)^ 2\oplus H^{n+2} F.\end{multlined} \] \[ \begin{multlined} S^ n: H^{n+1} M\oplus (H^{n+1} F)^ 3 \leftarrow \bigoplus^ 3_{i=1} H^{n+1} L_ i \leftarrow (H^{n+1} F)^ 2\leftarrow H^ n F\overset\text{cor}\longleftarrow\\ \overset \text{cor} \longleftarrow H^ n M\leftarrow \bigoplus^ 3_{i=1} H^ n L_ i\leftarrow H^ n M\oplus (H^ n F)^ 2\oplus H^{n-1} F.\end{multlined} \] Let \({\mathcal H}_ n(1)\), \({\mathcal H}_ n(2)\), \({\mathcal H}_ n(3)\), \({\mathcal H}_ n(4)\), and \({\mathcal H}_ n(5)\) be the homology groups of the complex \(S_ n\), and let \({\mathcal H}^ n(1)\), \({\mathcal H}^ n(2)\), \({\mathcal H}^ n(3)\), \({\mathcal H}^ n(4)\), and \({\mathcal H}^ n(5)\) be the homology groups of the complex \(S^ n\), corresponding to the terms of these complexes in the order presented above. It is shown that \({\mathcal H}_ n(2)\simeq {\mathcal H}^ n(2)\simeq {\mathcal H}_ n(4)\simeq {\mathcal H}^ n(4)\) and \({\mathcal H}_ n(3)\simeq {\mathcal H}^ n(3)\). Moreover, if \({\mathcal H}_ n(3)= {\mathcal H}^ n(3)=0\), then \({\mathcal H}_ n(1)={\mathcal H}^ n(1)={\mathcal H}_ n(5)={\mathcal H}^ n(5)=0\). Consequently, if \({\mathcal H}_ n(2)=0={\mathcal H}_ n(3)\) then \(S_ n\) and \(S^ n\) are exact sequences. It is readily shown that \({\mathcal H}_ n(2)=0={\mathcal H}_ n(3)\) for \(n=0\) or 1. To this point the paper is straightforward. The authors also show that \({\mathcal H}_ 2(2)=0={\mathcal H}_ 2(3)\) and \({\mathcal H}_ 3(3)=0\); but for this rather deep result, results about Galois cohomology, \(K\)-theory and Witt rings of quadratic forms are needed.
scientific article; zbMATH DE number 218291