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
is a ''purely inseparable extension'' if and only if for every
, the minimal polynomial of
over ''F'' is ''not'' a
separable polynomial.
[Isaacs, p. 298] 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