Editing Perfect field

{{Short description|Algebraic structure}}
In [[abstract algebra|algebra]], a [[field (mathematics)|field]] ''k'' is '''perfect''' if any one of the following equivalent conditions holds:
* Every [[irreducible polynomial]] over ''k'' has no [[Multiplicity (mathematics)#Multiplicity of a root of a polynomial|multiple roots]] in any [[field extension]] ''F/k''.
* Every [[irreducible polynomial]] over ''k'' has non-zero [[formal derivative]].
* 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 {{nowrap|''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 {{nowrap|''p'' > 0}}, the [[Frobenius endomorphism]] {{nowrap|''x'' ↦ ''x''{{i sup|''p''}}}} 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).

Another important property of perfect fields is that they admit [[Witt vector]]s.

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> (When restricted to [[integral domain]]s, this is equivalent to the above condition "every element of ''k'' is a ''p''th power".)

==Examples==

Examples of perfect fields are: 
* every field of characteristic zero, so <math>\mathbb{Q}</math> and every finite extension, and <math>\mathbb{C}</math>;<ref>Examples of fields of characteristic zero include the field of [[rational numbers]], the field of [[real numbers]] or the field of [[complex numbers]].</ref>
* every [[finite field]] <math>\mathbb{F}_q</math>;<ref>Any finite field of order ''q'' may be denoted <math>\mathbf{F}_{q}</math>, where ''q'' = ''p''{{i sup|''k''}} for some [[Prime number|prime]] ''p'' and [[positive integer]] ''k''.</ref>
* every [[algebraically closed field]];
* the union of a set of perfect fields [[totally ordered]] by extension;
* fields algebraic over a perfect field.

Most fields that are encountered in practice are perfect. The imperfect case arises mainly in [[algebraic geometry]] in characteristic {{nowrap|''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>\mathbf{F}_p(x)</math> of rational polynomials in an unknown element <math>x</math>. This can be seen from the fact that the Frobenius endomorphism sends <math>x \mapsto x^p</math> and therefore is not surjective. Equivalently, one can show that the polynomial <math>f(X)=X^p-x</math>, which is an element of <math>(\mathbf{F}_p(x))[X] </math>, is irreducible but inseparable.

This field embeds into the perfect field
:<math>\mathbf{F}_q(x,x^{1/p},x^{1/p^2},\ldots)</math>

called its '''perfection'''. Imperfect fields cause technical difficulties because irreducible polynomials can become reducible in the algebraic closure of the base field. For example,<ref>{{cite book|last1=Milne|first1=James|authorlink = James Milne (mathematician)|title=Elliptic Curves|pages=6|url=https://www.jmilne.org/math/Books/ectext6.pdf}}</ref> consider <math>f(x,y) = x^p + ay^p \in k[x,y]</math> for <math>k</math> an imperfect field of characteristic <math>p</math> and ''a'' not a ''p''-th power in ''k''. Then in its algebraic closure <math>k^{\operatorname{alg}}[x,y]</math>, the following equality holds:
:<math>
f(x,y) = (x + b y)^p ,
</math>
where ''b''{{i sup|''p''}} = ''a'' and such ''b'' exists in this algebraic closure. Geometrically, this means that <math>f</math> does not define an [[affine curve|affine]] [[plane curve]] in <math>k[x,y]</math>.

== Field extension over a perfect field ==
Any [[finitely generated field extension]] ''K'' over a perfect field ''k'' is separably generated, i.e. admits a separating [[transcendence basis|transcendence base]], that is, a transcendence base Γ such that ''K'' is separably algebraic over ''k''(Γ).<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''{{i sup|''r''}}-th roots ({{nowrap|''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]] {{nowrap|''u'' : ''A'' → ''A<sub>p</sub>''}} such that for any other perfect ring ''B'' of characteristic ''p'' with a homomorphism {{nowrap|''v'' : ''A'' → ''B''}} there is a unique homomorphism {{nowrap|''f'' : ''A<sub>p</sub>'' → ''B''}} such that ''v'' factors through ''u'' (i.e. {{nowrap|1=''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 {{nowrap|''θ'' : ''R''(''A'') → ''A''}} such that for any perfect ring ''B'' of characteristic ''p'' equipped with a map {{nowrap|''φ'' : ''B'' → ''A''}}, there is a unique map {{nowrap|''f'' : ''B'' → ''R''(''A'')}} such that ''φ'' factors through ''θ'' (i.e. {{nowrap|1=''φ'' = ''θf''}}). 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 {{nowrap|''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]]
*[[Perfect ring]]
*[[Quasi-finite field]]

==Notes==
{{reflist}}

== References ==
{{refbegin}}
*{{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=Cohn
|first =P.M.|authorlink = Paul Cohn
|year=2003
|title=Basic Algebra: Groups, Rings and Fields
}}
*{{Lang Algebra | edition=3r}}
*{{citation
|last=Matsumura
|first =Hideyuki|authorlink = Hideyuki Matsumura
|year=2003
|title=Commutative ring theory
|series=Translated from the Japanese by M. Reid. [[Cambridge Studies in Advanced Mathematics]]
|volume=8
|edition=2nd
}}
*{{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
}}
{{refend}}

==External links==
* {{springer|title=Perfect field|id=p/p072040}}

[[Category:Ring theory]]
[[Category:Field (mathematics)]]