Definition 1

Let G be a finite group, and let L be a nonzero commutative algebra of finite degree over K endowed with the structure of a $(K,G)$-algebra, where the action of G is given by an injective homomorphism from G to the group Aut$_K(L)$. We say that L is a Galois algebra with group G if it has the equivalent properties (i) through (vi) of Theorem 2.