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.
Read more about this topic: Separable Extension
Famous quotes containing the words purely, inseparable and/or extensions:
“Whats past and whats to come is strewed with husks
And formless ruin of oblivion;
But in this extant moment, faith and truth,
Strained purely from all hollow bias-drawing,
Bids thee, with most divine integrity,
From heart of very heart, great Hector, welcome!”
—William Shakespeare (15641616)
“Art is for [the Irish] inseparable from artifice: of that, the theatre is the home. Possibly, it was England made me a novelist.”
—Elizabeth Bowen (18991973)
“The psychological umbilical cord is more difficult to cut than the real one. We experience our children as extensions of ourselves, and we feel as though their behavior is an expression of something within us...instead of an expression of something in them. We see in our children our own reflection, and when we dont like what we see, we feel angry at the reflection.”
—Elaine Heffner (20th century)