Crystal system - Revision history

en>YiFeiBot: Bot: Migrating interwiki links, now provided by Wikidata on d:q898786

2014-11-17T16:47:59Z

Bot: Migrating interwiki links, now provided by Wikidata on d:q898786

← Older revision		Revision as of 18:47, 17 November 2014
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	In [[mathematics]], a '''Galois extension''' is an [[Algebraic extension\|algebraic field extension]] ''E''/''F'' that is [[normal extension\|normal]] and [[separable extension\|separable]]; or equivalently, ''E''/''F'' is [[algebraic extension\|algebraic]], and the [[fixed field\|field fixed]] by the [[automorphism]] group Aut(''E''/''F'') is precisely the base field ''F''. One says that such extension is '''Galois'''. The significance of being a Galois extension is that the extension has a [[Galois group]] and obeys the [[fundamental theorem of Galois theory]]. <ref> See the article [[Galois group]] for definitions of some of these terms and some examples.</ref>

	A result of [[Emil Artin]] allows one to construct Galois extensions as follows: If ''E'' is a given field, and ''G'' is a finite group of automorphisms of ''E'', then ''E''/''F'' is a Galois extension, where ''F'' is the fixed field of ''G''.

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	~~An important theorem of [[Emil Artin]] states that for a [[finite extension]] ''E''/'~~'F'~~', each of the following statements~~ is ~~equivalent to~~ the ~~statement~~ that ''E''/''~~F'' is Galois:~~
	* ''E''/''F'' ~~is a [[normal extension]]~~ and a ~~[[separable extension]]~~.
	* ''E'' is ~~a [[splitting field]]~~ of a ~~[[separable polynomial]] with coefficients in ''F''~~.
	* [''E'':''F''] = \|Aut(''E''/''F'')\|; that is, the ~~[[degree (field theory)\|degree]] of the field extension is equal~~ to ~~the [[order (group theory)\|order]] of the automorphism group of ''E''/''F'~~'.

	~~==Examples==~~

	~~[[Adjunction (field theory)\|Adjoining]]~~ to the ~~[[rational number field]] the square root~~ of ~~2 gives~~ a ~~Galois extension, while adjoining the cube root~~ of ~~2 gives a non-Galois extension. Both these extensions are separable, because they have [[characteristic zero]]~~. The ~~first of them~~ is ~~the splitting field of ''X''~~<~~sup~~>2<~~/sup~~> ~~− 2; the second has [[Normal extension\|normal closure]]~~ that ~~includes the complex~~ [~~[cube roots of unity~~]~~], and so is not~~ a ~~splitting field~~. ~~In fact~~, it ~~has no automorphism other than the identity~~, ~~because it~~ is ~~contained~~ in the ~~real numbers and ''X''<sup>3</sup> − 2 has just one real root~~.

	~~An [[algebraic closure]]~~ <~~math~~>~~\bar K~~<~~/math~~> of ~~an arbitrary field <math>K</math>~~ is ~~Galois over <math>K</math> if~~ and ~~only if~~ <~~math~~>K<~~/math~~> is a ~~[[perfect field]].~~

	~~== See also ==~~
	* {{cite book \| author=Emil Artin \| title=Galois Theory \| publisher=Dover Publications \| year=1998 \| isbn=0-486-62342-4 \| authorlink=Emil Artin}} ''(Reprinting of ~~second revised edition of 1944~~, ~~The University of Notre Dame Press)''~~.
	* {{cite book \| author=[[Jörg Bewersdorff]] \| title=Galois Theory for Beginners: A Historical Perspective\| publisher=American Mathematical Society \| year=2006 \| isbn=0-8218-3817-2}} .
	* {{cite book\|author=Harold M. Edwards \| authorlink = Harold Edwards (mathematician)\| title=Galois Theory \| publisher=Springer-Verlag \| year = 1984 \| isbn=0-387-90980-X}} ''(Galois' original paper, ~~with extensive background~~ and ~~commentary~~.~~)''~~
	* {{cite journal\| first= H. Gray \| last=Funkhouser \| title=A ~~short account~~ of ~~the history~~ of ~~symmetric functions of roots of equations \| journal=American Mathematical Monthly \| year=1930 \| volume= 37 \| issue=7 \| pages=357–365 \| doi=10~~.~~2307/2299273\| publisher= The American Mathematical Monthly, Vol. 37, No. 7\| ref= harv\| jstor= 2299273 }}~~
	* {{springer\|title=Galois theory\|id=p/g043160}}
	* {{cite book \| author=Nathan Jacobson\| title=Basic Algebra I (2nd ed) \| publisher=W.H. Freeman and ~~Company \| year=1985 \| isbn=0-7167-1480-9 \| authorlink=Nathan Jacobson}} ''(Chapter 4 gives an introduction to the field-theoretic approach~~ to ~~Galois theory~~.~~)''~~
	* {{Cite book \| last1=Janelidze \| first1=G. \| last2=Borceux \| first2=Francis \| title=Galois theories \| publisher=[[Cambridge University Press]] \| isbn=978-0-521-80309-0 \| year=2001 \| ref=harv \| postscript=<!~~--None--~~>~~}} (This book introduces~~ the ~~reader~~ to the ~~Galois theory~~ of ~~[[Grothendieck]]~~, and ~~some generalisations, leading~~ to ~~Galois [[groupoids]]~~.)
	* {{Cite book \| last1=Lang \| first1=Serge \| author1-link=Serge Lang \| title=Algebraic Number Theory \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| isbn=978-0-387-94225-4 \| year=1994 \| ref=harv \| postscript=<!--None-->}}
	* {{cite book\|author=M. M. Postnikov \| title=Foundations of ~~Galois Theory \| publisher=Dover Publications \| year = 2004 \| isbn=0-486-43518-0}}~~
	* {{cite book\|author=Joseph Rotman \| title =Galois Theory (~~2nd edition)\| publisher=Springer\| year=1998 \| isbn=0-387-98541-7}}~~
	* {{Cite book \| last1=Völklein \| first1=Helmut \| title=Groups as Galois groups: an introduction \| publisher=[~~[Cambridge University Press]] \| isbn=978-0-521-56280-5 \| year=1996 \| ref=harv \| postscript=<!--None-->}}~~
	* {{Cite book \| last1=van der Waerden \| first1=Bartel Leendert \| author1-link=Bartel Leendert van der Waerden \| title=Moderne Algebra \|language= German \|publisher=Springer \| year=1931 \| location=Berlin \|ref=harv \| postscript=<!--None-->}}. '''English translation''' (of 2nd revised edition): {{Cite book \| title = Modern algebra \| publisher=Frederick Ungar \|location= New York \|year= 1949}} ''(Later republished in English by Springer under the title "Algebra".)''
	* {{Cite web \|title=(Some) New Trends in Galois Theory and Arithmetic \|first=Florian \|last=Pop \|authorlink=Florian Pop\|url=http://~~www~~.~~math.upenn~~.~~edu/~pop/Research/files-Res/Japan01.pdf \|year=2001 \|ref=harv \|postscript=<!--None-->}}~~

	~~== References ==~~
	~~<references />~~

	~~{{DEFAULTSORT:Galois Extension}}~~
	~~[[Category:Galois theory]]~~
	~~[[Category:Algebraic number theory]]~~
	~~[[Category:Field extensions]]~~

en>Tomruen: /* Lattice systems */

2014-01-03T20:55:07Z

Lattice systems

← Older revision		Revision as of 22:55, 3 January 2014
Line 1:		Line 1:
	~~Today I am sharing~~ a ~~dark experience with you as I am from this darkness now~~. ~~Whoa! Google Chrome~~ has ~~crashed was~~ the ~~disaster which struck on me early this morning~~. ~~My blood flow appeared to stop~~ for ~~a limited minutes. Next I regained my senses~~ and ~~decided to fix Chrome crash~~.<br>~~<br>Windows Defender - this does come standard with several Windows OS Machines~~, ~~nevertheless otherwise~~ is ~~download from Microsoft~~ for ~~free~~. ~~It can aid safeguard against spyware~~.~~<br><br>So what if you look for when we compare registry products~~. ~~Many~~ of the ~~registry cleaners accessible now~~, have ~~fairly similar features~~. The ~~leading ones which you need to be seeking are these.~~<br><br>~~The problem with nearly all~~ of ~~the persons~~ is ~~that they do~~ not ~~desire to spend money~~. In the ~~cracked variation one refuses to have to pay anything plus can download it from web easily. It is easy to install as well. However~~, ~~the problem comes whenever~~ it ~~really~~ is ~~not able to identify all possible viruses, spyware plus malware inside~~ the ~~system~~. ~~This really~~ is ~~considering it happens to be obsolete inside nature~~ and ~~does not get any usual changes from the site downloaded. Thus, a system is accessible to difficulties like hacking.~~<br><br>~~There are~~ a ~~lot~~ of [~~http~~:~~//bestregistrycleanerfix~~.~~com/tune~~-up-~~utilities tuneup utilities 2014] s~~. ~~Which 1 is the greatest is not convenient to be determined~~. ~~But if we wish To stand out one among~~ the ~~multitude we should consider some items~~. ~~These are features~~, ~~scanning speed time~~, ~~total mistakes detected, total mistakes repaired, tech support, Boot time performance~~ and ~~cost~~. ~~According to these products Top Registry Cleaner for 2010 is RegCure~~.<br>~~<br>Turn It Off: Chances are when you are like me; then you spend a great deal of time on a computer on a daily basis. Try providing a computer several time~~ to ~~do absolutely nothing; this will sound funny however in~~ the ~~event you have an elder computer you may be asking it~~ to ~~do too much~~.<~~br><br~~>~~Most probably when you are experiencing a slow computer it may be a couple years old~~. ~~You additionally will not have been told that whilst we utilize your computer everyday; there are certain aspects that it requires to continue running in its best performance~~. You furthermore will not even own any diagnostic tools which can get your PC running like modern again. So do not let which stop you from getting your system cleaned. With access to the web you will find the tools that will help you get your system running like brand-~~new again.~~<br><br>~~Another important program you~~'~~ll wish To receive is a registry cleaner. The registry is a huge list~~ of ~~everything installed on your computer, plus Windows references it when it opens a system or utilizes a device connected to your computer~~. ~~Whenever you delete a system, its registry entry could furthermore be deleted, nevertheless occasionally it~~'~~s not~~. ~~A registry cleaner can get rid of these aged entries so Windows may search the registry faster~~. ~~It furthermore deletes or corrects any entries which viruses have corrupted~~.		In [[mathematics]], a '''Galois extension''' is an [[Algebraic extension\|algebraic field extension]] ''E''/''F'' that is [[normal extension\|normal]] and [[separable extension\|separable]]; or equivalently, ''E''/''F'' is [[algebraic extension\|algebraic]], and the [[fixed field\|field fixed]] by the [[automorphism]] group Aut(''E''/''F'') is precisely the base field ''F''. One says that such extension is '''Galois'''. The significance of being a Galois extension is that the extension has a [[Galois group]] and obeys the [[fundamental theorem of Galois theory]]. <ref> See the article [[Galois group]] for definitions of some of these terms and some examples.</ref>

			A result of [[Emil Artin]] allows one to construct Galois extensions as follows: If ''E'' is a given field, and ''G'' is a finite group of automorphisms of ''E'', then ''E''/''F'' is a Galois extension, where ''F'' is the fixed field of ''G''.

			==Characterization of Galois extensions==
			An important theorem of [[Emil Artin]] states that for a [[finite extension]] ''E''/''F'', each of the following statements is equivalent to the statement that ''E''/''F'' is Galois:
			* ''E''/''F'' is a [[normal extension]] and a [[separable extension]].
			* ''E'' is a [[splitting field]] of a [[separable polynomial]] with coefficients in ''F''.
			* [''E'':''F''] = \|Aut(''E''/''F'')\|; that is, the [[degree (field theory)\|degree]] of the field extension is equal to the [[order (group theory)\|order]] of the automorphism group of ''E''/''F''.

			==Examples==

			[[Adjunction (field theory)\|Adjoining]] to the [[rational number field]] the square root of 2 gives a Galois extension, while adjoining the cube root of 2 gives a non-Galois extension. Both these extensions are separable, because they have [[characteristic zero]]. The first of them is the splitting field of ''X''<sup>2</sup> − 2; the second has [[Normal extension\|normal closure]] that includes the complex [[cube roots of unity]], and so is not a splitting field. In fact, it has no automorphism other than the identity, because it is contained in the real numbers and ''X''<sup>3</sup> − 2 has just one real root.

			An [[algebraic closure]] <math>\bar K</math> of an arbitrary field <math>K</math> is Galois over <math>K</math> if and only if <math>K</math> is a [[perfect field]].

			== See also ==
			* {{cite book \| author=Emil Artin \| title=Galois Theory \| publisher=Dover Publications \| year=1998 \| isbn=0-486-62342-4 \| authorlink=Emil Artin}} ''(Reprinting of second revised edition of 1944, The University of Notre Dame Press)''.
			* {{cite book \| author=[[Jörg Bewersdorff]] \| title=Galois Theory for Beginners: A Historical Perspective\| publisher=American Mathematical Society \| year=2006 \| isbn=0-8218-3817-2}} .
			* {{cite book\|author=Harold M. Edwards \| authorlink = Harold Edwards (mathematician)\| title=Galois Theory \| publisher=Springer-Verlag \| year = 1984 \| isbn=0-387-90980-X}} ''(Galois' original paper, with extensive background and commentary.)''
			* {{cite journal\| first= H. Gray \| last=Funkhouser \| title=A short account of the history of symmetric functions of roots of equations \| journal=American Mathematical Monthly \| year=1930 \| volume= 37 \| issue=7 \| pages=357–365 \| doi=10.2307/2299273\| publisher= The American Mathematical Monthly, Vol. 37, No. 7\| ref= harv\| jstor= 2299273 }}
			* {{springer\|title=Galois theory\|id=p/g043160}}
			* {{cite book \| author=Nathan Jacobson\| title=Basic Algebra I (2nd ed) \| publisher=W.H. Freeman and Company \| year=1985 \| isbn=0-7167-1480-9 \| authorlink=Nathan Jacobson}} ''(Chapter 4 gives an introduction to the field-theoretic approach to Galois theory.)''
			* {{Cite book \| last1=Janelidze \| first1=G. \| last2=Borceux \| first2=Francis \| title=Galois theories \| publisher=[[Cambridge University Press]] \| isbn=978-0-521-80309-0 \| year=2001 \| ref=harv \| postscript=<!--None-->}} (This book introduces the reader to the Galois theory of [[Grothendieck]], and some generalisations, leading to Galois [[groupoids]].)
			* {{Cite book \| last1=Lang \| first1=Serge \| author1-link=Serge Lang \| title=Algebraic Number Theory \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| isbn=978-0-387-94225-4 \| year=1994 \| ref=harv \| postscript=<!--None-->}}
			* {{cite book\|author=M. M. Postnikov \| title=Foundations of Galois Theory \| publisher=Dover Publications \| year = 2004 \| isbn=0-486-43518-0}}
			* {{cite book\|author=Joseph Rotman \| title =Galois Theory (2nd edition)\| publisher=Springer\| year=1998 \| isbn=0-387-98541-7}}
			* {{Cite book \| last1=Völklein \| first1=Helmut \| title=Groups as Galois groups: an introduction \| publisher=[[Cambridge University Press]] \| isbn=978-0-521-56280-5 \| year=1996 \| ref=harv \| postscript=<!--None-->}}
			* {{Cite book \| last1=van der Waerden \| first1=Bartel Leendert \| author1-link=Bartel Leendert van der Waerden \| title=Moderne Algebra \|language= German \|publisher=Springer \| year=1931 \| location=Berlin \|ref=harv \| postscript=<!--None-->}}. '''English translation''' (of 2nd revised edition): {{Cite book \| title = Modern algebra \| publisher=Frederick Ungar \|location= New York \|year= 1949}} ''(Later republished in English by Springer under the title "Algebra".)''
			* {{Cite web \|title=(Some) New Trends in Galois Theory and Arithmetic \|first=Florian \|last=Pop \|authorlink=Florian Pop\|url=http://www.math.upenn.edu/~pop/Research/files-Res/Japan01.pdf \|year=2001 \|ref=harv \|postscript=<!--None-->}}

			== References ==
			<references />

			{{DEFAULTSORT:Galois Extension}}
			[[Category:Galois theory]]
			[[Category:Algebraic number theory]]
			[[Category:Field extensions]]

24.6.0.130: /* Crystal classes */ fixed wording to restrict to chiral biological molecules

2012-07-23T01:02:31Z

Crystal classes: fixed wording to restrict to chiral biological molecules

New page

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