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Separable extension

In mathematics, an algebraic field extension L/K is separable if it can be generated by adjoining to K a set each of whose elements 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 (equivalently, algebraic) extensions are separable. There exists a simple criterion for perfectness: a field F is perfect if and only if

• F has characteristic 0, or
• F has a nonzero characteristic p, and every element of F has a p-th root in F.

Equivalently, the second condition says that the Frobenius endomorphism of F, $xmapsto x^p$, is an automorphism.

In particular, 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 largest subfield M of L containing K such that M is a separable extension of K. When M = K the extension L/K is called a purely inseparable extension. In general an algebraic extension factors as a purely inseparable extension of a separable extension, since the compositum of a family of separable extensions is again separable.

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 Ksep inside an algebraic closure, which is the largest separable subextension of Kalg/K. Then Galois theory can be carried out inside Ksep.

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