perfect field

A perfect field is a field $K$ such that every algebraic extension field $L/K$ is separable over $K$.

All fields of characteristic 0 are perfect, so in particular the fields $ℝ$, $ℂ$ and $\mathbb{Q}$ are perfect. If $K$ is a field of characteristic $p$ (with $p$ a prime number), then $K$ is perfect if and only if the Frobenius endomorphism $F$ on $K$, defined by

$$F(x)=x^p\mathit{\hspace{1em}}(x\in K),$$

is an automorphism of $K$. Since the Frobenius map is always injective, it is sufficient to check whether $F$ is surjective. In particular, all finite fields are perfect (any injective endomorphism is also surjective). Moreover, any field whose characteristic is nonzero that is algebraic (http://planetmath.org/AlgebraicExtension) over its prime subfield 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 $x\mapsto x^p$ 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