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 $\mathbb{R}$, $\mathbb{C}$ 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

 $F(x)=x^{p}\quad(x\in K),$

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 PerfectField 2013-03-22 13:08:23 2013-03-22 13:08:23 sleske (997) sleske (997) 11 sleske (997) Definition msc 12F10 SeparablePolynomial ExtensionField perfect perfect ring