Characterization of Finite Galois Extensions

Summary In this article we prove the well-known characterization of finite Galois extensions: a finite extension E of F is a Galois extension of F i E is both normal and separable i E is the splitting field of a separable polynomial p ∈ F [ X ]. We also prove some applications of the characterization, so for example that F ( a 1 , . . ., a n ) is a separable extension of F if and only if all the a i are separable, or that every finite separable extension of F is contained in a Galois extension of F .