every algebraically closed field is perfect

Proposition 1.

Every algebraically closed field is perfectPlanetmathPlanetmath


Let K be an algebraically closed field of prime characteristicPlanetmathPlanetmath p. Take aK. Then the polynomialPlanetmathPlanetmath Xp-a admits a zero in K. It follows that a admits a pth root in K. Since a is arbitrary we have proved that the field K is perfect.∎

Title every algebraically closed field is perfect
Canonical name EveryAlgebraicallyClosedFieldIsPerfect
Date of creation 2013-03-22 16:53:06
Last modified on 2013-03-22 16:53:06
Owner polarbear (3475)
Last modified by polarbear (3475)
Numerical id 6
Author polarbear (3475)
Entry type Result
Classification msc 12F05