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perfect field
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(Definition)
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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 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).
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"perfect field" is owned by sleske. [ full author list (5) ]
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Cross-references: ring, transcendental, prime subfield, finite fields, surjective, sufficient, injective, Frobenius map, automorphism, Frobenius endomorphism, prime number, characteristic, separable, algebraic extension, field
There are 14 references to this entry.
This is version 8 of perfect field, born on 2002-11-08, modified 2008-03-12.
Object id is 3577, canonical name is PerfectField.
Accessed 7751 times total.
Classification:
| AMS MSC: | 12F10 (Field theory and polynomials :: Field extensions :: Separable extensions, Galois theory) |
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Pending Errata and Addenda
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