perfect field
A perfect field is a field such that every algebraic extension
field is separable
over .
All fields of characteristic 0 are perfect, so in particular the fields , and are perfect. If is a field of characteristic (with a prime number), then is perfect if and only if the Frobenius endomorphism on , defined by
is an automorphism of . Since the Frobenius map is always injective
, it is sufficient to check whether 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 of characteristic is perfect if the endomorphism of 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 |