
Purely inseparable extension

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 "x""q" = "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.
Purely inseparable extensions.
An algebraic extension formula_1 is a "purely inseparable extension" if and only if for every formula_2, the minimal polynomial of formula_3 over "F" is "not" a separable polynomial. If "F" is any field, the trivial extension formula_4 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 formula_1 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 formula_6, there exists formula_7 such that formula_8.
3. Each element of "E" has minimal polynomial over "F" of the form formula_9 for some integer formula_7 and some element formula_11.
It follows from the above equivalent characterizations that if formula_12 (for "F" a field of prime characteristic) such that formula_8 for some integer formula_7, then "E" is purely inseparable over "F". (To see this, note that the set of all "x" such that formula_15 for some formula_7 forms a field; since this field contains both formula_3 and "F", it must be "E", and by condition 2 above, formula_1 must be purely inseparable.)
If "F" is an imperfect field of prime characteristic "p", choose formula_11 such that "a" is not a "p"th power in "F", and let "f"("X") = "X"p − "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 formula_3 with formula_21. In particular, formula_22 and by the property stated in the paragraph directly above, it follows that formula_23 is a non-trivial purely inseparable extension (in fact, formula_12, and so formula_1 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 "p"th 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".
Galois correspondence for purely inseparable extensions.
 introduced a variation of Galois theory for purely inseparable extensions of exponent 1, where the Galois groups of field automorphisms in Galois theory are replaced by restricted Lie algebras of derivations. The simplest case is for finite index purely inseparable extensions "K"⊆"L" of exponent at most 1 (meaning that the "p"th power of every element of "L" is in "K"). In this case the Lie algebra of "K"-derivations of "L" is a restricted Lie algebra that is also a vector space of dimension "n" over "L", where ["L":"K"] = "p""n", and the intermediate fields in "L" containing "K" correspond to the restricted Lie subalgebras of this Lie algebra that are vector spaces over "L". Although the Lie algebra of derivations is a vector space over "L", it is not in general a Lie algebra over "L", but is a Lie algebra over "K" of dimension "n"["L":"K"] = "np""n".
A purely inseparable extension is called a modular extension if it is a tensor product of simple extensions, so in particular every extension of exponent 1 is modular, but there are non-modular extensions of exponent 2 .
 and gave an extension of the Galois correspondence to modular purely inseparable extensions, where derivations are replaced by higher derivations.