Separable extension

In field theory, a branch of algebra, an algebraic field extension ${\displaystyle E/F}$ is called a separable extension if 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 the zero polynomial, or equivalently it has no repeated roots in any extension field).^[1] There is also a more general definition that applies when E is not necessarily algebraic over F. An extension that is not separable is said to be inseparable.

Every algebraic extension of a field of characteristic zero is separable, and every algebraic extension of a finite field is separable.^[2] 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 assumed to be separable.^[3]

The opposite concept, a purely inseparable extension, also occurs naturally, as every algebraic extension may be decomposed uniquely as a purely inseparable extension of a separable extension. An algebraic extension ${\displaystyle E/F}$ of fields of non-zero characteristic p is a purely inseparable extension if and only if for every ${\displaystyle \alpha \in E\setminus F}$, the minimal polynomial of ${\displaystyle \alpha }$ over F is not a separable polynomial, or, equivalently, for every element x of E, there is a positive integer k such that ${\displaystyle x^{p^{k}}\in F}$.^[4]

The simplest nontrivial example of a (purely) inseparable extension is ${\displaystyle E=\mathbb {F} _{p}(x)\supseteq F=\mathbb {F} _{p}(x^{p})}$, fields of rational functions in the indeterminate x with coefficients in the finite field ${\displaystyle \mathbb {F} _{p}=\mathbb {Z} /(p)}$. The element ${\displaystyle x\in E}$ has minimal polynomial ${\displaystyle f(X)=X^{p}-x^{p}\in F[X]}$, having ${\displaystyle f'(X)=0}$ and a p-fold multiple root, as ${\displaystyle f(X)=(X-x)^{p}\in E[X]}$. This is a simple algebraic extension of degree p, as ${\displaystyle E=F[x]}$, but it is not a normal extension since the Galois group ${\displaystyle {\text{Gal}}(E/F)}$ is trivial.