Separable Extension - Purely Inseparable Extensions

Purely Inseparable Extensions

An algebraic extension is a purely inseparable extension if and only if for every, the minimal polynomial of over F is not a separable polynomial. If F is any field, the trivial extension is purely inseparable; for the field F to possess a non-trivial purely inseparable extension, it must be imperfect as outlined in the above section.

Several equivalent and more concrete definitions for the notion of a purely inseparable extension are known. If is an algebraic extension with (non-zero) prime characteristic p, then the following are equivalent:

1. E is purely inseparable over F.

2. For each element, there exists such that .

3. Each element of E has minimal polynomial over F of the form for some integer and some element .

It follows from the above equivalent characterizations that if (for F a field of prime characteristic) such that for some integer, then E is purely inseparable over F. (To see this, note that the set of all x such that for some forms a field; since this field contains both and F, it must be E, and by condition 2 above, must be purely inseparable.)

If F is an imperfect field of prime characteristic p, choose such that a is not a pth power in F, and let f(X)=Xp−a. Then f has no root in F, and so if E is a splitting field for f over F, it is possible to choose with . In particular, and by the property stated in the paragraph directly above, it follows that is a non-trivial purely inseparable extension (in fact, and so is automatically a purely inseparable extension).

Purely inseparable extensions do occur naturally; for example, they occur in algebraic geometry over fields of prime characteristic. If K is a field of characteristic p, and if V is an algebraic variety over K of dimension greater than zero, the function field K(V) is a purely inseparable extension over the subfield K(V)p of pth powers (this follows from condition 2 above). Such extensions occur in the context of multiplication by p on an elliptic curve over a finite field of characteristic p.