Characterization of Finite Galois Extensions

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 (a₁, . . ., 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 .