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Concept
Purely inseparable extension
Summary
In algebra, a purely inseparable extension of fields is an extension k ⊆ K of fields of characteristic p > 0 such that every element of K is a root of an equation of the form xq = a, with q a power of p and a in k. Purely inseparable extensions are sometimes called radicial extensions, which should not be confused with the similar-sounding but more general notion of radical 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:
E is purely inseparable over F.
For each element , there exists such that .
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.
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