Inseparable Extension

An inseparable extension is an algebraic field extension which is not separable. Thus some element has an extension field minimal polynomial with a multiple root in an algebraic closure. Inseparable extensions can occur only in positive field characteristic; for example, adjoining a th root of to the rational function field gives an inseparable extension.

An inseparable extension need not be a purely inseparable extension. More precisely, for an algebraic extension , the elements of separable over form an intermediate field . Then is a separable extension and is a purely inseparable extension. The extension field is inseparable iff , and it is purely inseparable iff (Stacks Project, Tag 09HD).