The perfect closure of a field 
of characteristic 
, inside a fixed algebraic
closure, is
 |
where 
consists of the elements whose 
th powers lie in 
. It is a purely
inseparable extension and the smallest perfect field
containing 
in that closure. In characteristic zero,
the perfect closure is 
itself.
The Frobenius map is an automorphism of 
,
so integer powers with negative exponents are also defined.
Passing to this closure removes the purely inseparable ambiguity in the function
field reconstruction assertions using prime-to-characteristic
invariants.
