A purely inseparable extension is an algebraic extension 
over a field 
such that the extension
field minimal polynomial over 
of every element of 
has only one distinct root
in an algebraic closure of 
, possibly with multiplicity.
In positive field characteristic 
, this is equivalent to requiring that for every 
there is a nonnegative
integer 
with 
.
The integer 
may depend on 
. A finite extension
that is purely inseparable has extension field
degree a power of 
, since it can be built by a finite tower of extensions
of extension field degree 
. In characteristic
zero, the only purely inseparable extension is 
. More generally, an extension
field is both purely inseparable and a separable
extension iff it is trivial (Stacks Project, Tag 09HD).
