An
algebraic extension E\supseteq F is a
purely inseparable extension if and only if for every \alpha\in E\setminus F, the
minimal polynomial of \alpha over
F is
not a
separable polynomial. If
F is any field, the trivial extension F\supseteq F 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 E\supseteq F is an algebraic extension with (non-zero) prime characteristic
p, then the following are equivalent: •
E is purely inseparable over
F. • For each element \alpha\in E, there exists n\geq 0 such that \alpha^{p^n}\in F. • Each element of
E has minimal polynomial over
F of the form X^{p^n}-a for some integer n\geq 0 and some element a\in F. It follows from the above equivalent characterizations that if E=F[\alpha] (for
F a field of prime characteristic) such that \alpha^{p^n}\in F for some integer n\geq 0, then
E is purely inseparable over
F. (To see this, note that the set of all
x such that x^{p^n}\in F for some n\geq 0 forms a field; since this field contains both \alpha and
F, it must be
E, and by condition 2 above, E\supseteq F must be purely inseparable.) If
F is an imperfect field of prime characteristic
p, choose a\in F 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 \alpha with f(\alpha)=0. In particular, \alpha^{p}=a and by the property stated in the paragraph directly above, it follows that F[\alpha]\supseteq F is a non-trivial purely inseparable extension (in fact, E=F[\alpha], and so E\supseteq F is automatically a purely inseparable extension). Purely inseparable extensions do occur naturally; for example, 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.
Properties • If the characteristic of a field
F is a (non-zero) prime number
p, and if E\supseteq F is a purely inseparable extension, then if F\subseteq K\subseteq E,
K is purely inseparable over
F and
E is purely inseparable over
K. Furthermore, if [
E :
F] is finite, then it is a power of
p, the characteristic of
F. • Conversely, if F\subseteq K\subseteq E is such that F\subseteq K and K\subseteq E are purely inseparable extensions, then
E is purely inseparable over
F. • An algebraic extension E\supseteq F is an
inseparable extension if and only if there is
some \alpha\in E\setminus F such that the minimal polynomial of \alpha over
F is
not a
separable polynomial (i.e., an algebraic extension is inseparable if and only if it is not separable; note, however, that an inseparable extension is not the same thing as a purely inseparable extension). If E\supseteq F is a finite degree non-trivial inseparable extension, then [
E :
F] is necessarily divisible by the characteristic of
F. • If E\supseteq F is a finite degree normal extension, and if K=\mbox{Fix}(\mbox{Gal}(E/F)), then
K is purely inseparable over
F and
E is separable over
K. ==Galois correspondence for purely inseparable extensions==