Separable extension

In field theory, a subfield of algebra, a separable extension is an algebraic field extension ${\displaystyle E\supset F}$ such that for every ${\displaystyle \alpha \in E}$, the minimal polynomial of ${\displaystyle \alpha }$ over F is a separable polynomial (i.e., its formal derivative is not zero; see below for other equivalent definitions). Otherwise, the extension is said to be inseparable.

In characteristic zero, every algebraic extension is separable, and every algebraic extension of a finite field is separable. It follows that most extensions that are considered in mathematics are separable. Nevertheless, the concept of separability is important, as the existence of inseparable extensions is the main obstacle for extending many theorems proved in characteristic zero to non-zero characteristic. For example, the fundamental theorem of Galois theory is a theorem about normal extensions, which remains true in non-zero characteristic only if the extensions are also supposed to be separable.

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