Dust solution - Revision history

en>Colonies Chris: sp, date & link fixes; unlinking common words, replaced: stress-energy tensor → stress–energy tensor using AWB

2014-03-13T17:00:18Z

sp, date & link fixes; unlinking common words, replaced: stress-energy tensor → stress–energy tensor using AWB

← Older revision		Revision as of 19:00, 13 March 2014
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	~~:''This article~~ is ~~about~~ the ~~Hauptidealsatz of [[class field theory]]~~. ~~You may be seeking [[Krull~~'s ~~principal ideal theorem]], also known as Krull's Hauptidealsatz,~~ in ~~[[commutative algebra]]''~~		Hi there. My name is Sophia Meagher even though it is not the name on my beginning certificate. Invoicing is my profession. For years he's been residing in Mississippi and he psychic love readings - [http://myoceancounty.net/groups/apply-these-guidelines-when-gardening-and-grow/ link webpage] - doesn't plan on altering it. The favorite pastime for him and his phone psychic ([https://www.machlitim.org.il/subdomain/megila/end/node/12300 www.machlitim.org.il]) kids is to play lacross and he would by no means give it up.<br><br>Look at my blog post; free online tarot card readings ([http://afeen.fbho.net/v2/index.php?do=/profile-210/info/ More Signup bonuses])

	~~In [[mathematics]], the '''principal ideal theorem''' of [[class field theory]], a branch of [[algebraic number theory]], is the statement that for any [[algebraic number field]] ''K''~~ and ~~any ideal ''I'' of the [[ring of integers]] of ''K'', if ''L'' is the~~ [~~[Hilbert class field]] of ''K'', then~~

	:~~<math>IO_L\ <~~/~~math>~~

	~~is a [[principal ideal]] α''O''<sub>''L''<~~/~~sub>, for ''O''<sub>''L''<~~/~~sub> the ring of integers of ''L''~~ and ~~some element α in it. In other terms, extending ideals gives a mapping on the [[class group~~]~~] of~~ '~~'K'', to the class group of ''L'', which sends all ideal classes to the class of a principal ideal~~. The ~~phenomenon has also been called ''principalization'', or sometimes ''capitulation''. It was conjectured by [[David Hilbert]],~~ and ~~was the last remaining aspect of~~ his ~~programme on class fields to be completed, around 1930.~~

	~~The question was reduced to a piece of finite group theory by [[Emil Artin]]. That involved the [[transfer~~ (~~group theory)\|transfer]]. The required result was proved by~~ [~~[Philipp Furtwängler]]~~.

	~~==References==~~
	* {{cite journal \| first=Philipp \| last=Furtwängler \| authorlink=Philipp Furtwängler \| title=Beweis des Hauptidealsatzes fur Klassenkörper algebraischer Zahlkörper \| journal=Abh. ~~Math~~. ~~Sem~~. ~~Hamburg \| volume=7 \| year=1929 \| pages=14–36 \| jfm=55~~.~~0699~~.~~02 }}~~
	* {{cite book \| last=Gras \| first=Georges \| title=Class field theory. From theory to ~~practice \| series=Springer Monographs in Mathematics \| location=Berlin \| publisher=[[Springer-Verlag]] \| year=2003 \| isbn=3-540-44133-6 \| zbl=1019~~.~~11032 }}~~
	* {{cite book \| first=Helmut \| last=Koch \| title=Algebraic Number Theory \| publisher=[~~[Springer-Verlag]] \| year=1997 \| isbn=3-540-63003-1 \| zbl=0819~~.~~11044 \| series=Encycl~~. ~~Math. Sci~~. ~~\| volume~~=~~62 \| edition=2nd printing of 1st \| page=104 }}~~
	*{{cite book \| last=Serre \| first=Jean-Pierre \| authorlink=Jean-Pierre Serre \| title=Local fields \| others=Translated from the French by Marvin Jay Greenberg \| series=[[Graduate Texts in Mathematics]] \| volume=67 \| publisher=[[Springer-Verlag]] \| year=1979 \| isbn=0-387-90424-7 \| zbl=0423.12016 \| pages=120–122 }}

	~~[[Category:Ideals]]~~
	~~[[Category:Group theory]]~~
	~~[[Category:Homological algebra]]~~
	~~[[Category:Theorems in algebraic number theory]~~]

en>Rep07: Rm POV adjective "fascinating".

2013-11-12T05:46:39Z

Rm POV adjective "fascinating".

← Older revision		Revision as of 07:46, 12 November 2013
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	~~Greetings~~. ~~The author~~'s ~~title~~ is ~~Phebe~~ and ~~she feels comfy when people use~~ the ~~complete name~~. ~~To do aerobics is~~ a ~~factor that I~~'~~m totally addicted~~ to~~. My working day job is~~ a ~~librarian~~. ~~Minnesota~~ has ~~always~~ been his ~~house but his wife wants them~~ to transfer.~~<br><br>Here is my blog~~ - [~~http://kitakeluarga~~.~~com/index~~.~~php?do~~=~~/profile~~-~~12707/info/ http~~:~~//kitakeluarga.com/~~]		:''This article is about the Hauptidealsatz of [[class field theory]]. You may be seeking [[Krull's principal ideal theorem]], also known as Krull's Hauptidealsatz, in [[commutative algebra]]''

			In [[mathematics]], the '''principal ideal theorem''' of [[class field theory]], a branch of [[algebraic number theory]], is the statement that for any [[algebraic number field]] ''K'' and any ideal ''I'' of the [[ring of integers]] of ''K'', if ''L'' is the [[Hilbert class field]] of ''K'', then

			:<math>IO_L\ </math>

			is a [[principal ideal]] α''O''<sub>''L''</sub>, for ''O''<sub>''L''</sub> the ring of integers of ''L'' and some element α in it. In other terms, extending ideals gives a mapping on the [[class group]] of ''K'', to the class group of ''L'', which sends all ideal classes to the class of a principal ideal. The phenomenon has also been called ''principalization'', or sometimes ''capitulation''. It was conjectured by [[David Hilbert]], and was the last remaining aspect of his programme on class fields to be completed, around 1930.

			The question was reduced to a piece of finite group theory by [[Emil Artin]]. That involved the [[transfer (group theory)\|transfer]]. The required result was proved by [[Philipp Furtwängler]].

			==References==
			* {{cite journal \| first=Philipp \| last=Furtwängler \| authorlink=Philipp Furtwängler \| title=Beweis des Hauptidealsatzes fur Klassenkörper algebraischer Zahlkörper \| journal=Abh. Math. Sem. Hamburg \| volume=7 \| year=1929 \| pages=14–36 \| jfm=55.0699.02 }}
			* {{cite book \| last=Gras \| first=Georges \| title=Class field theory. From theory to practice \| series=Springer Monographs in Mathematics \| location=Berlin \| publisher=[[Springer-Verlag]] \| year=2003 \| isbn=3-540-44133-6 \| zbl=1019.11032 }}
			* {{cite book \| first=Helmut \| last=Koch \| title=Algebraic Number Theory \| publisher=[[Springer-Verlag]] \| year=1997 \| isbn=3-540-63003-1 \| zbl=0819.11044 \| series=Encycl. Math. Sci. \| volume=62 \| edition=2nd printing of 1st \| page=104 }}
			*{{cite book \| last=Serre \| first=Jean-Pierre \| authorlink=Jean-Pierre Serre \| title=Local fields \| others=Translated from the French by Marvin Jay Greenberg \| series=[[Graduate Texts in Mathematics]] \| volume=67 \| publisher=[[Springer-Verlag]] \| year=1979 \| isbn=0-387-90424-7 \| zbl=0423.12016 \| pages=120–122 }}

			[[Category:Ideals]]
			[[Category:Group theory]]
			[[Category:Homological algebra]]
			[[Category:Theorems in algebraic number theory]]

en>Phn229: /* Examples */

2012-04-07T02:31:58Z

Examples

New page

Greetings. The author's title is Phebe and she feels comfy when people use the complete name. To do aerobics is a factor that I'm totally addicted to. My working day job is a librarian. Minnesota has always been his house but his wife wants them to transfer.<br><br>Here is my blog - [http://kitakeluarga.com/index.php?do=/profile-12707/info/ http://kitakeluarga.com/]