Definition 17.7.1 (Splitting field). A splitting field of a polynomial is an extension such that factors completely into linear factors in and is generated over by the roots of . A splitting field exists for any and is unique up to (non-unique) isomorphism.
Definition 17.7.2 (Normal extension). A finite extension is normal if every irreducible polynomial in that has one root in splits completely in . Equivalently, is a splitting field of some polynomial over .
Definition 17.7.3 (Separable extension). A polynomial is separable if it has no repeated roots (in its splitting field). An algebraic element is separable over if its minimal polynomial is separable. An algebraic extension is separable if every element of is separable over . In characteristic 0, every algebraic extension is separable.