# Separable extension

In mathematics, a **separable extension** of a field *K *is a field *L* containing *K* that can be generated by adjoining to *K* a set of elements α, each of which is a root of a separable polynomial over *K*. In that case, each β in *L* has a separable minimal polynomial over *K*.

The condition of separability is central in Galois theory. A **perfect field** is one for which all finite extensions are separable.

Then

- all fields of characteristic 0, and
- all finite fields,

are perfect. This means that the separability condition can be assumed in many contexts. The effects of inseparability (necessarily for infinite *K* of characteristic *p*) can be seen in the primitive element theorem, and for the tensor product of fields.

Given a finite extension *L*/*K* of fields, there is a smallest subfield *M* of *L* containing *K* such that *L* is a separable extension of *M*. When *L* = *M* the extension *L*/*K* is called a **purely inseparable extension**. In general finite extensions factor as a separable extension of a purely inseparable extension, because a separable extension of a separable extension is again a separable extension.

Purely inseparable extensions do occur for quite natural reasons, for example in algebraic geometry in characteristic *p*. If *K* is a field of characteristic *p*, and *V* an algebraic variety over *K* of dimension > 0, consider the function field *K*(*V*) and its subfield *K*(*V*)^{p} of *p*-th powers. This is always a purely inseparable extension. Such extensions occur as soon as one looks at multiplication by *p* on an elliptic curve over a finite field of characteristic *p*.

In dealing with non-perfect fields *K*, one introduces the **separable closure** *K*^{sep} inside an algebraic closure, which is the largest available separable extension of *K*. Then Galois theory can be carried out inside *K*^{sep}.