Twin sextic algebras (Q1288331)

Let \(K\) be a separable sextic algebra over a ground field \(F\). Then \(K\) has a `` twin'' \(K^t\), well-defined up to unique isomorphism, determined by a nontrivial outer isomorphism \(t\) of the symmetric group \(S_6\). This paper is concerned with a number of topics related to this ` twinning'\ operation, and of considerable use in investigating sextic algebras over a given ground field. The paper begins with a useful review of Galois theory from a categorical viewpoint, then proceeds to describe all possible Galois groups of sextic algebras. The exceptional outer automorphism \(t\) is analyzed and is used to describe the conjugacy classes and characters of the Galois groups of sextic algebras. The paper concludes with some geometric examples.