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.