On a Hasse principle for \(\sigma\)-conjugacy (Q916714)

Let G be an algebraic group defined over the number field K such that \(G(K)=A^{\times}\), where A is a semisimple algebra over K. If M is a cyclic extension of K with generator \(\sigma\) of the Galois group one studies the \(\sigma\)-conjugacy classes in G(M) (i.e., g and \(g'\) belong to the same class if \(g'=h^{-1}g^{\sigma}h\) for some h). It is shown that \(\sigma\)-conjugacy classes are mapped bijectively by a natural norm map to the \(\sigma\)-invariant conjugacy classes which are locally everywhere in the image of that map. As a consequence one sees that two elements of G(M) are \(\sigma\)-conjugate if and only if they are \(\sigma\)-conjugate locally everywhere.