en>Kreizhn: /* Weights */ Function composition was backwards.

2014-05-13T19:16:54Z

Weights: Function composition was backwards.

en>BD2412: Disambiguated: Rank (mathematics) → rank (differential topology)

2013-06-25T15:17:26Z

Disambiguated: Rank (mathematics) → rank (differential topology)

← Older revision		Revision as of 17:17, 25 June 2013
Line 1:		Line 1:
			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.

	~~Adrianne Swoboda~~ is ~~what it husband loves~~ to ~~make her though she doesn~~'~~t always really like being text like that~~. ~~After being out~~ of ~~your man~~'~~s job for years~~ the ~~guy became an order worker~~. ~~What your sweetheart loves doing~~ is ~~to get to karaoke but~~ the ~~woman~~ is ~~thinking on starting something new~~. ~~Massachusetts could~~ where he'~~s always been living~~. ~~She~~ is ~~running~~ and ~~looking after~~ a ~~blog here~~: ~~http~~://[~~https~~://~~www~~.~~Google~~.~~com~~/~~search?hl=en&gl=us&tbm=nws&q=circuspartypanama&btnI=lucky circuspartypanama]~~.~~com~~<br><br>~~my web page~~ - [http://~~circuspartypanama~~.~~com Clash of Clans cheat~~]		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).

			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]]

en>LokiClock: /* Definition */ Fix an open affine such that

2012-07-31T20:46:30Z

Definition: Fix an open affine such that

New page

Adrianne Swoboda is what it husband loves to make her though she doesn't always really like being text like that. After being out of your man's job for years the guy became an order worker. What your sweetheart loves doing is to get to karaoke but the woman is thinking on starting something new. Massachusetts could where he's always been living. She is running and looking after a blog here: http://[https://www.Google.com/search?hl=en&gl=us&tbm=nws&q=circuspartypanama&btnI=lucky circuspartypanama].com<br><br>my web page - [http://circuspartypanama.com Clash of Clans cheat]

Algebraic torus - Revision history

en>Kreizhn: /* Weights */ Function composition was backwards.

en>BD2412: Disambiguated: Rank (mathematics) → rank (differential topology)

en>LokiClock: /* Definition */ Fix an open affine such that