Oscillatory integral - Revision history

en>Rjwilmsi: /* References */Added 1 doi to a journal cite using AWB (10209)

2014-05-23T11:39:58Z

References: Added 1 doi to a journal cite using AWB (10209)

← Older revision		Revision as of 13:39, 23 May 2014
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	~~{{Confusing\|article\|date=February 2010}}~~		Hello and welcome. My title is Figures Wunder. My day occupation is a meter reader. One of the extremely best things in the world for me is to do aerobics and now I'm attempting to earn money with it. For many years he's been residing in North Dakota and his family enjoys it.<br><br>Feel free to visit my web blog [http://www.sddch.org/?document_srl=345265 over the counter std test]
	~~:''For reflection principles in set theory, see [[reflection principle]].''~~

	~~In [[algebraic number theory]], a '''reflection theorem''' or '''Spiegelungssatz''' ([[German language\|German]] for ''reflection theorem'' – see ''[[Spiegel (disambiguation)\|Spiegel]]''~~ and ''[[Satz (disambiguation)\|Satz]]'') is one of a collection of theorems linking the sizes of different [[ideal class group]]s (or [[ray class group]]s), or the sizes of different [[isotypic component]]s of a class group. The original example is due to [[Ernst Kummer\|Ernst Eduard Kummer]], who showed that the class number of the [[cyclotomic field]] <math>\mathbb{Q} \left( \zeta_p \right)</math>, with ''p'' a prime number, will be divisible by ''p'' if the class number of the maximal real subfield <math>\mathbb{Q} \left( \zeta_p \right)^{+}</math> is. ~~Another example~~ is due to Scholz.<ref>A. Scholz, Uber die Beziehung der Klassenzahlen quadratischer Korper zueinander, ''J. reine angew. Math.'', '''166''' (1932), 201-203.</ref> A simplified version of his theorem states that if 3 divides the class number of a ~~[[real quadratic field]] <math>\mathbb{Q} \left( \sqrt{d} \right)</math>, then 3 also divides the class number of the [[imaginary quadratic field]] <math>\mathbb{Q} \left( \sqrt{-3d} \right)</math>~~.

	~~==Leopoldt's Spiegelungssatz==~~
	~~Both~~ of the ~~above results are generalized by [[Heinrich-Wolfgang Leopoldt\|Leopoldt]]'s "Spiegelungssatz", which relates~~ the ~~[[p-rank]]s of different isotypic components of the class group of a number field considered as a [[Galois module\|module]] over the [[Galois group]] of a Galois extension.~~

	~~Let ''L''/''K'' be a finite Galois extension of number fields, with group ''G'', degree prime~~ to ~~''p''~~ and '~~'L'' containing the ''p''-th roots of unity~~. ~~Let~~ '~~'A'' be the ''p''-Sylow subgroup of the class group of ''L''. Let φ run over the irreducible characters of the group ring '''Q'''<sub>''p''</sub>[''G'']~~ and ~~let ''A''<sub>φ</sub> denote the corresponding direct summands of ''A''~~. ~~For any φ let ''q'' = ''p''~~<~~sup~~>~~φ(1)~~<~~/sub> and let the ''G''-rank ''e''<sub>φ</sub> be the exponent in the index~~

	~~:<math~~> [ ~~A_\phi : A_\phi^p ] = q^{e_\phi} . </math>~~

	~~Let ω be the character of ''G''~~

	:~~<math> \zeta^g = \zeta^{\omega(g)} \text{ for } \zeta \in \mu_p . </math>~~

	~~The reflection (''Spiegelung'') φ<sup>*</sup> is defined by~~

	~~:<math> \phi^*(g) = \omega(g) \phi(g^{-1}) . </math>~~

	Let ''E'' be the unit group of ''K''. We say that ε is "primary" if <math>K(\sqrt[p]\epsilon)/K</math> is unramified, and let ''E''<sub>0</sub> denote the group of primary units modulo ''E''<sup>''p''</sup>. Let δ<sub>φ</~~sub> denote the ''G''-rank of the φ component of ''E''<sub>0<~~/~~sub>~~.

	~~The Spiegelungssatz states that~~

	~~:<math> \| e_{\phi^*} - e_\phi \| \le \delta_\phi~~ . </~~math>~~

	~~==Extensions=~~=
	Extensions of this Spiegelungssatz were given by Oriat and Oriat-Satge, where class groups were no longer associated with characters of the Galois group of ''K''/''k'', but rather by ideals in a [[group ring]] over the Galois group of ''K''/''k''. Leopoldt's Spiegelungssatz was generalized in a different direction by Kuroda, who extended it to a statement about [[ray class group]]s. This was further developed into the very general "''T''-''S'' reflection theorem" of [[Georges Gras]].<ref>Georges Gras, ''Class Field Theory: From Theory to Practice'', Springer-Verlag, Berlin, 2004, pp. 157–158.</ref> [[Kenkichi Iwasawa]] also provided an [[Iwasawa theory\|Iwasawa-theoretic]] reflection theorem.

	~~==References==~~
	~~{{reflist}}~~
	* {{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 \| pages=147–149 }}

	~~[[Category:Theorems in algebraic number theory]~~]

en>Myasuda: /* Definition */ avoid redirect

2013-11-02T14:46:45Z

Definition: avoid redirect

← Older revision		Revision as of 16:46, 2 November 2013
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	The ~~name~~ of the ~~author~~ is ~~Numbers~~. ~~His wife doesn~~'~~t like it~~ the ~~way he does but what he truly likes performing is~~ to ~~do aerobics~~ and he'~~s been doing it for fairly a while~~. ~~Her family members lives~~ in ~~Minnesota~~. ~~My day occupation is a librarian~~.<br><br>~~Here~~ is ~~my website~~ - [~~http:~~//~~www~~.~~hurtdating~~.~~com~~/groups/~~candida~~-~~albicans~~-~~tips~~-~~you~~-~~have~~-to-~~remember/ home std test~~]		{{Confusing\|article\|date=February 2010}}
			:''For reflection principles in set theory, see [[reflection principle]].''

			In [[algebraic number theory]], a '''reflection theorem''' or '''Spiegelungssatz''' ([[German language\|German]] for ''reflection theorem'' – see ''[[Spiegel (disambiguation)\|Spiegel]]'' and ''[[Satz (disambiguation)\|Satz]]'') is one of a collection of theorems linking the sizes of different [[ideal class group]]s (or [[ray class group]]s), or the sizes of different [[isotypic component]]s of a class group. The original example is due to [[Ernst Kummer\|Ernst Eduard Kummer]], who showed that the class number of the [[cyclotomic field]] <math>\mathbb{Q} \left( \zeta_p \right)</math>, with ''p'' a prime number, will be divisible by ''p'' if the class number of the maximal real subfield <math>\mathbb{Q} \left( \zeta_p \right)^{+}</math> is. Another example is due to Scholz.<ref>A. Scholz, Uber die Beziehung der Klassenzahlen quadratischer Korper zueinander, ''J. reine angew. Math.'', '''166''' (1932), 201-203.</ref> A simplified version of his theorem states that if 3 divides the class number of a [[real quadratic field]] <math>\mathbb{Q} \left( \sqrt{d} \right)</math>, then 3 also divides the class number of the [[imaginary quadratic field]] <math>\mathbb{Q} \left( \sqrt{-3d} \right)</math>.

			==Leopoldt's Spiegelungssatz==
			Both of the above results are generalized by [[Heinrich-Wolfgang Leopoldt\|Leopoldt]]'s "Spiegelungssatz", which relates the [[p-rank]]s of different isotypic components of the class group of a number field considered as a [[Galois module\|module]] over the [[Galois group]] of a Galois extension.

			Let ''L''/''K'' be a finite Galois extension of number fields, with group ''G'', degree prime to ''p'' and ''L'' containing the ''p''-th roots of unity. Let ''A'' be the ''p''-Sylow subgroup of the class group of ''L''. Let φ run over the irreducible characters of the group ring '''Q'''<sub>''p''</sub>[''G''] and let ''A''<sub>φ</sub> denote the corresponding direct summands of ''A''. For any φ let ''q'' = ''p''<sup>φ(1)</sub> and let the ''G''-rank ''e''<sub>φ</sub> be the exponent in the index

			:<math> [ A_\phi : A_\phi^p ] = q^{e_\phi} . </math>

			Let ω be the character of ''G''

			:<math> \zeta^g = \zeta^{\omega(g)} \text{ for } \zeta \in \mu_p . </math>

			The reflection (''Spiegelung'') φ<sup>*</sup> is defined by

			:<math> \phi^*(g) = \omega(g) \phi(g^{-1}) . </math>

			Let ''E'' be the unit group of ''K''. We say that ε is "primary" if <math>K(\sqrt[p]\epsilon)/K</math> is unramified, and let ''E''<sub>0</sub> denote the group of primary units modulo ''E''<sup>''p''</sup>. Let δ<sub>φ</sub> denote the ''G''-rank of the φ component of ''E''<sub>0</sub>.

			The Spiegelungssatz states that

			:<math> \| e_{\phi^*} - e_\phi \| \le \delta_\phi . </math>

			==Extensions==
			Extensions of this Spiegelungssatz were given by Oriat and Oriat-Satge, where class groups were no longer associated with characters of the Galois group of ''K''/''k'', but rather by ideals in a [[group ring]] over the Galois group of ''K''/''k''. Leopoldt's Spiegelungssatz was generalized in a different direction by Kuroda, who extended it to a statement about [[ray class group]]s. This was further developed into the very general "''T''-''S'' reflection theorem" of [[Georges Gras]].<ref>Georges Gras, ''Class Field Theory: From Theory to Practice'', Springer-Verlag, Berlin, 2004, pp. 157–158.</ref> [[Kenkichi Iwasawa]] also provided an [[Iwasawa theory\|Iwasawa-theoretic]] reflection theorem.

			==References==
			{{reflist}}
			* {{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 \| pages=147–149 }}

			[[Category:Theorems in algebraic number theory]]

67.188.231.139 at 22:09, 5 February 2012

2012-02-05T22:09:59Z

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The name of the author is Numbers. His wife doesn't like it the way he does but what he truly likes performing is to do aerobics and he's been doing it for fairly a while. Her family members lives in Minnesota. My day occupation is a librarian.<br><br>Here is my website - [http://www.hurtdating.com/groups/candida-albicans-tips-you-have-to-remember/ home std test]