The Classical Galois Theorem

This first chapter presents the classical Galois theorem for fields. A finite-dimensional extension of fields K ⊆ L is a Galois extension when every element l ∈ L is root of a polynomial p(X) ∈ K[X] which factors in L[X] into distinct linear factors. The Galois group Gal[L : K] of that extension is the group of all field endomorphisms (and thus automorphisms) of L which fix all the elements of K. The Galois theorem exhibits a bijection between the subgroups of the Galois group and the intermediate field extensions K ⊆ M ⊆ L.