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In [[algebra]], a [[field (mathematics)|field]] ''k'' is said to be '''perfect''' if any one of the following equivalent conditions holds:
* Every [[irreducible polynomial]] over ''k'' has distinct roots.
* Every [[irreducible polynomial]] over ''k'' is [[separable polynomial|separable]].
* Every [[finite extension]] of ''k'' is [[separable extension|separable]].
* Every [[algebraic extension]] of ''k'' is separable.
* Either ''k'' has [[characteristic (algebra)|characteristic]] 0, or, when ''k'' has characteristic ''p'' > 0, every element of ''k'' is a [[Power (mathematics)|''p''th power]].
* Either ''k'' has [[characteristic (algebra)|characteristic]] 0, or, when ''k'' has characteristic ''p'' > 0, the [[Frobenius endomorphism]] ''x''→''x''<sup>''p''</sup> is an [[automorphism]] of ''k''
* The [[separable closure]] of ''k'' is [[algebraically closed]].
* Every [[reduced ring|reduced]] commutative [[Algebra (ring theory)|''k''-algebra]] ''A'' is a [[separable algebra]]; i.e., <math>A \otimes_k F</math> is [[reduced ring|reduced]] for every [[field extension]] ''F''/''k''. (see below)
Otherwise, ''k'' is called '''imperfect'''.


In particular, all fields of characteristic zero and all [[finite field]]s are perfect.


Perfect fields are significant because [[Galois theory]] over these fields becomes simpler, since the general Galois assumption of field extensions being separable is automatically satisfied over these fields (see third condition above).
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More generally, a [[ring (mathematics)|ring]] of characteristic ''p'' (''p'' a [[prime number|prime]]) is called '''perfect''' if the [[Frobenius endomorphism]] is an [[automorphism]].<ref>{{harvnb|Serre|1979}}, Section II.4</ref> (This is equivalent to the above condition "every element of ''k'' is a ''p''th power" for integral domains.)
 
==Examples==
 
Examples of perfect fields are:
* every field of characteristic zero, e.g. the field of [[rational numbers]] or the field of [[complex numbers]];
* every [[finite field]], e.g. the field '''F'''<sub>''p''</sub> = '''Z'''/''p'''''Z''' where ''p'' is a [[prime number]];
* every [[algebraically closed field]];
* the union of a set of perfect fields totally ordered by extension;
* fields algebraic over a perfect field.
 
In fact, most fields that appear in practice are perfect. The imperfect case arises mainly in algebraic geometry in characteristic ''p''>0. Every imperfect field is necessarily [[Field_extension#Algebraic_and_transcendental_elements|transcendental]] over its [[prime subfield]] (the minimal subfield), because the latter is perfect. An example of an imperfect field is
*the field <math>k(X)</math> of all rational functions in an indeterminate <math>X</math>, where ''k'' has characteristic ''p''>0 (because ''X'' has no ''p''-th root in ''k''(''X'')).
 
== Field extension over a perfect field ==
Any [[finitely generated field extension]] over a perfect field is [[separably generated field extension|separably generated]].<ref>Matsumura, Theorem 26.2</ref>
 
==Perfect closure and perfection==
One of the equivalent conditions says that, in characteristic ''p'', a field adjoined with all ''p''<sup>''r''</sup>-th roots (''r''≥1) is perfect; it is called the '''perfect closure''' of ''k'' and usually denoted by <math>k^{p^{-\infty}}</math>.
 
The perfect closure can be used in a test for separability. More precisely, a commutative ''k''-algebra ''A'' is separable if and only if <math>A \otimes_k k^{p^{-\infty}}</math> is reduced.<ref>{{harvnb|Cohn|2003|loc=Theorem 11.6.10}}</ref>
 
In terms of [[universal property|universal properties]], the '''perfect closure''' of a ring ''A'' of characteristic ''p'' is a perfect ring ''A<sub>p</sub>'' of characteristic ''p'' together with a [[ring homomorphism]] ''u'' : ''A'' → ''A<sub>p</sub>'' such that for any other perfect ring ''B'' of characteristic ''p'' with a homomorphism ''v'' : ''A'' → ''B'' there is a unique homomorphism ''f'' : ''A<sub>p</sub>'' → ''B'' such that ''v'' factors through ''u'' (i.e. ''v'' = ''fu''). The perfect closure always exists; the proof involves "adjoining ''p''-th roots of elements of ''A''", similar to the case of fields.<ref>{{harvnb|Bourbaki|2003}}, Section V.5.1.4, page 111</ref>
 
The '''perfection''' of a ring ''A'' of characteristic ''p'' is the dual notion (though this term is sometimes used for the perfect closure). In other words, the perfection ''R''(''A'') of ''A'' is a perfect ring of characteristic ''p'' together with a map θ : ''R''(''A'') → ''A'' such that for any perfect ring ''B'' of characteristic ''p'' equipped with a map φ : ''B'' → ''A'', there is a unique map ''f'' : ''B'' → ''R''(''A'') such that φ factors through θ (i.e. φ = θ''f''). The perfection of ''A'' may be constructed as follows. Consider the [[projective system]]
:<math>\cdots\rightarrow A\rightarrow A\rightarrow A\rightarrow\cdots</math>
where the transition maps are the Frobenius endomorphism. The [[inverse limit]] of this system is ''R''(''A'') and consists of sequences (''x''<sub>0</sub>, ''x''<sub>1</sub>, ... ) of elements of ''A'' such that <math>x_{i+1}^p=x_i</math> for all ''i''. The map θ : ''R''(''A'') → ''A'' sends (''x<sub>i</sub>'') to ''x''<sub>0</sub>.<ref>{{harvnb|Brinon|Conrad|2009}}, section 4.2</ref>
 
== See also ==
*[[p-ring]]
*[[Quasi-finite field]]
 
==Notes==
{{reflist}}
 
== References ==
*{{Citation
| last=Bourbaki
| first=Nicolas
| author-link=Nicolas Bourbaki
| title=Algebra II
| isbn=978-3-540-00706-7
| publisher=Springer
| year=2003
}}
*{{Citation
| last=Brinon
| first=Olivier
| last2=Conrad
| first2=Brian
| author2-link=Brian Conrad
| title=CMI Summer School notes on p-adic Hodge theory
| url=http://math.stanford.edu/~conrad/papers/notes.pdf
| year=2009
| accessdate=2010-02-05
}}
*{{Citation
| last=Serre
| first=Jean-Pierre
| author-link=Jean-Pierre Serre
| title=[[Local Fields (book)|Local fields]]
| year=1979
| edition=2
| publisher=[[Springer-Verlag]]
| series=[[Graduate Texts in Mathematics]]
| volume=67
| mr=554237
| isbn=978-0-387-90424-5
}}
*{{citation
|last=Cohn
|first =P.M.
|year=2003
|title=Basic Algebra: Groups, Rings and Fields
}}
*{{citation
|last=Matsumura
|first =H
|year=2003
|title=Commutative ring theory
|series=Translated from the Japanese by M. Reid. [[Cambridge Studies in Advanced Mathematics]]
|volume=8
|edition=2nd
}}
 
==External links==
* {{springer|title=Perfect field|id=p/p072040}}
 
[[Category:Ring theory]]
[[Category:Field theory]]

Latest revision as of 21:16, 13 May 2014


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