Separability

From Definition 1.1, checking a polynomial is separable requires building a splitting field to check the roots are distinct. But we will see in Section 2 a criterion for deciding when a polynomial is separable (that is, has no multiple roots) without having to work in a splitting field. In Section 3 we will define what it means for a field extension to be separable and then prove the primitive element theorem, an important result about separable field extensions.