Abelian crossed products and scalar extensions of central simple algebras

Let $\mathcal A$ be a central simple algebra over a field $k$ and $G$ a commutative group with $|G|=\deg(\mathcal A)$. We prove that there exists a regular field extension $E/k$ preserving indices of $k$-algebras such that $\mathcal A\otimes_k E$ is a crossed product with the group $G$. Bibl. – 11 titles.