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.
In positive field characteristic, 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.
