Separable Extension

In modern algebra, an algebraic field extension is a separable extension if and only if for every, the minimal polynomial of over F is a separable polynomial (i.e., has distinct roots). Otherwise, the extension is called inseparable. There are other equivalent definitions of the notion of a separable algebraic extension, and these are outlined later in the article.

The class of separable extensions is an extremely important one due to the fundamental role it plays in Galois theory. More specifically, a finite degree field extension is Galois if and only if it is both normal and separable. Since algebraic extensions of fields of characteristic zero, and of finite fields, are separable, separability is not an obstacle in most applications of Galois theory. For instance, every algebraic (in particular, finite degree) extension of the field of rational numbers is necessarily separable.

Despite the ubiquity of the class of separable extensions in mathematics, its extreme opposite, namely the class of purely inseparable extensions, also occurs quite naturally. An algebraic extension is a purely inseparable extension if and only if for every, the minimal polynomial of over F is not a separable polynomial (i.e., does not have distinct roots). For a field F to possess a non-trivial purely inseparable extension, it must necessarily be an infinite field of prime characteristic (i.e. specifically, imperfect), since any algebraic extension of a perfect field is necessarily separable.

The study of separable extensions in their own right has far-reaching consequences. For instance, consider the result: "If E is a field with the property that every nonconstant polynomial with coefficients in E has a root in E, then E is algebraically closed." Despite its simplicity, it suggests a deeper conjecture: "If is an algebraic extension and if every nonconstant polynomial with coefficients in F has a root in E, is E algebraically closed?" Although this conjecture is true, most of its known proofs depend on the theory of separable and purely inseparable extensions; for instance, in the case corresponding to the extension being separable, one known proof involves the use of the primitive element theorem in the context of Galois extensions.