perfect field

A perfect fieldMathworldPlanetmath is a field K such that every algebraic extensionMathworldPlanetmath field L/K is separablePlanetmathPlanetmath over K.

All fields of characteristic 0 are perfect, so in particular the fields , and are perfect. If K is a field of characteristic p (with p a prime numberMathworldPlanetmath), then K is perfect if and only if the Frobenius endomorphism F on K, defined by


is an automorphismPlanetmathPlanetmathPlanetmath of K. Since the Frobenius map is always injectivePlanetmathPlanetmath, it is sufficient to check whether F is surjectivePlanetmathPlanetmath. In particular, all finite fieldsMathworldPlanetmath are perfect (any injective endomorphism is also surjective). Moreover, any field whose characteristic is nonzero that is algebraicMathworldPlanetmath ( over its prime subfieldMathworldPlanetmath is perfect. Thus, the only fields that are not perfect are those whose characteristic is nonzero and are transcendental over their prime subfield.

Similarly, a ring R of characteristic p is perfect if the endomorphism xxp of R is an automorphism (i.e., is surjective).

Title perfect field
Canonical name PerfectField
Date of creation 2013-03-22 13:08:23
Last modified on 2013-03-22 13:08:23
Owner sleske (997)
Last modified by sleske (997)
Numerical id 11
Author sleske (997)
Entry type Definition
Classification msc 12F10
Related topic SeparablePolynomial
Related topic ExtensionField
Defines perfect
Defines perfect ring