Separable extension

From Example Problems
Jump to navigation Jump to search

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

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