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 In mathematics, a polynomial ''P''(''X'') over a given field ''K'' is separable if its roots are distinct in an algebraic closure of ''K'', that is, the number of distinct roots is equal to the degree of the polynomial.
This concept is closely ...
.
[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