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Algebraic Types Of Homomorphisms Necessary And Sufficient Ring Relation Theory Prime Field Characteristic Finite Field Algebraic Extension Separable Polynomial Prime Subfield Frobenius Automorphism Frobenius Map

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

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

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 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 $x\mapsto x^p$ of is an automorphism (i.e., is surjective).

1	Types Of Homomorphisms
1	Field
1	Relation Theory
1	Necessary And Sufficient
1	Algebraic
1	Ring
1	Prime
3	Characteristic
3	Finite Field
4	Separable Polynomial
4	Algebraic Extension
7	Prime Subfield
10	Frobenius Map
17	Frobenius Automorphism