Separable extension

of a field

An extension such that for some natural number the fields and are linearly disjoint over (see Linearly-disjoint extensions). An extension that is not separable is called inseparable. Here is the characteristic of . In characteristic 0 all extensions are separable.

In what follows only algebraic extensions will be considered (for transcendental separable extensions see Transcendental extension). A finite extension is separable if and only if the trace mapping is a non-zero function. An algebraic extension is separable if any finite subextension is separable.

The separable extensions form a distinguished class of extensions, that is, in a tower of fields the extension is separable if and only if and are separable; if and are separable extensions, then so is ; for a separable extension and an arbitrary extension the extension is again separable. An extension is separable if and only if it admits an imbedding in a Galois extension . In this case, the number of different -isomorphisms of into is the same as the degree for a finite extension . Any finite separable extension is simple.

A polynomial is called separable over if none of its irreducible factors has a multiple root in an algebraic closure of . An algebraic element is called separable (over ) if it is a root of a polynomial that is separable over . Otherwise is called inseparable. An element is called purely inseparable over if for some . An irreducible polynomial is inseparable if and only if its derivative is identically zero (this is possible only for of characteristic and ). An arbitrary irreducible polynomial can be uniquely represented in the form , where is a separable polynomial. The degree of and the number are called, respectively, the reduced degree and the index of .

Let be an arbitrary algebraic extension. The elements of the field that are separable over form a field, , which is the maximal separable extension of contained in . The field is called the separable closure of in . The degree is called the separable degree of , and the degree the inseparable degree, or the degree of inseparability. The inseparable degree is equal to some power of the number . If , then is said to be separably closed in . In this case the extension is called purely inseparable. An extension is purely inseparable if and only if

that is, if any element of is purely inseparable over . The purely inseparable extensions of a field form a distinguished class of extensions. If an extension is both separable and purely inseparable, then . For references see Extension of a field.