8. Separable Extensions

In this chapter, we discuss the last property of field extensions which is needed to define Galois extensions: separability (which is usually satisfied in most common situations). An extension of a field is separable if any irreducible polynomial with coefficients in this field does not have multiple zeros. All extensions of fields of characteristic zero and all finite extensions of finite fields have this property. For this reason, there are sometimes “simplified presentations” of Galois theory in which one studies only fields of characteristic zero and finite fields. It is then not necessary to mention separability and the theoretical background necessary for the main theorems of Galois theory is more modest. We choose to discuss separability here as nonseparable field extensions are important in many branches of mathematics. In this chapter, we characterize separable extensions and prove the theorem on primitive element which says that a finite separable extension can be generated over its ground field by only one element. This theorem is usually part of any standard course on the subject.