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| In [[number theory]], '''Iwasawa theory''' is the study of objects of arithmetic interest over infinite [[Tower of fields|towers]] of [[number field]]s. It began as a [[Galois module]] theory of [[ideal class group]]s, initiated by {{harvs|txt|authorlink=Kenkichi Iwasawa|last=Iwasawa|year=1959}}, as part of the theory of [[cyclotomic field]]s. In the early 1970s, [[Barry Mazur]] considered generalizations of Iwasawa theory to [[abelian variety|abelian varieties]]. More recently (early 90s), [[Ralph Greenberg]] has proposed an Iwasawa theory for [[motive (algebraic geometry)|motives]].
| | This Halloween, make a fresh custom. Instead of it being a day, or days, of indulging inside the favorite sugary treats, make it a day for creating healthy resolutions.<br><br>Now regarding calorie intake, the bulky bodybuilder must reduce it. What you need to do first is determine what a basal metabolic rate is (BMR) is. This really is the calorie amount a body burns at rest. There is a fast plus easy-to-use free [http://safedietplans.com/bmr-calculator bmr calculator] we can employ on my webpage. Just click on the link under this short article to check it out. Once you learn what the BMR is you need to take in at least 500 nevertheless no over 1000 calories lower than this number. Keeping the calorie expenditure inside this range may aid we burn off the fat without muscle together with it. Plus, you'll keep the metabolic rate significant for maximum fat reduction. This is how to receive ripped rapidly with calorie intake in the event you happe to be bulky.<br><br>Food items rich inside fibers: The food items that are especially quite rich in fibers will improve basal metabolic rate at a steady pace. Actually, the fiber content inside the food items would assist to process or break the food particles at a steady pace. Further, you are able to even add protein wealthy items to strengthen the body muscles. Acai berry, prunes plus grapes may additionally be further added in your diet chart.<br><br>If you're eating fewer calories than you're burning, you'll lose fat. How much? Well, consider it this way - by burning an extra 500 calories a day, you'll lose one pound every week. Slow progress, sure, however, it's a ideal illustration of how, by keeping track of what you're doing, you can succeed at losing weight.<br><br>The daily bmr plus the calories you burn for sport and different escapades make the total amount of calories you burn a day. This really is precisely why BMR is so significant - it helps you plan a weight loss or weight gain program.<br><br>Most immediate weight reduction diets don't include any form of bodily exercise. They motivate you to lose weight rapidly by eating pretty little. While which will lead to a rapid initial weight reduction, the fat usually come back when you start to eat usually again.<br><br>In summary, we can do a lot to better the metabolism. Lifestyle change plus consistent frequent exercise, all may improve metabolic rate. |
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| ==Formulation==
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| Iwasawa worked with so-called <math>\mathbb{Z}_p</math>-extensions: infinite extensions of a [[number field]] <math> F </math> with [[Galois group]] <math> \Gamma </math> isomorphic to the additive group of [[p-adic integer]]s for some prime ''p''. Every closed subgroup of <math> \Gamma </math> is of the form <math> \Gamma^{p^n} </math>, so by Galois theory, a <math> \mathbb{Z}_p </math>-extension <math> F_\infty/F </math> is the same thing as a tower of fields <math> F = F_0 \subset F_1 \subset F_2 \subset \ldots \subset F_\infty </math> such that <math>\textrm{Gal}(F_n/F)\cong \mathbb{Z}/p^n\mathbb{Z}</math>. Iwasawa studied classical Galois modules over <math> F_n </math> by asking questions about the structure of modules over <math>F_\infty</math>.
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| More generally, Iwasawa theory asks questions about the structure of Galois modules over extensions with Galois group a [[p-adic Lie group]].
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| ==Example==
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| Let ''p'' be a prime number and let ''K'' = '''Q'''(μ<sub>''p''</sub>) be the field generated over '''Q''' by the ''p''th roots of unity. Iwasawa considered the following tower of number fields:
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| :<math> K = K_{0} \subset K_{1} \subset \cdots \subset K_{\infty}, </math>
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| where <math>K_n</math> is the field generated by adjoining to <math>K</math> the ''p<sup>n''+1</sup>st roots of unity and <math> K_\infty = \bigcup K_n </math>. The fact that <math>\textrm{Gal}(K_n/K)\simeq \mathbb{Z}/p^n\mathbb{Z}</math> implies, by infinite Galois theory, that <math>\textrm{Gal}(K_{\infty}/K)</math> is isomorphic to <math> \mathbb{Z}_p </math>. In order to get an interesting Galois module here, Iwasawa took the ideal class group of <math>K_n</math>, and let <math>I_n</math> be its ''p''-torsion part. There are [[field norm|norm]] maps <math>I_m\rightarrow I_n</math> whenever <math>m>n</math>, and this gives us the data of an [[inverse limit|inverse system]]. If we set <math>I = \varprojlim I_n</math>, then it is not hard to see from the inverse limit construction that <math> I </math> is a module over <math> \mathbb{Z}_p</math>. In fact, <math>I</math> is a [[module (mathematics)|module]] over the [[Iwasawa algebra]] <math>\Lambda=\mathbb{Z}_p[[\Gamma]]</math>. This is a [[Krull dimension|2-dimensional]], [[regular local ring]], and this makes it possible to describe modules over it. From this description it is possible to recover information about the ''p''-part of the class group of <math> K</math>.
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| The motivation here is that the ''p''-torsion in the ideal class group of <math>K</math> had already been identified by [[Ernst Kummer|Kummer]] as the main obstruction to the direct proof of [[Fermat's last theorem]].
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| ==Connections with p-adic analysis==
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| From this beginning in the 1950s, a substantial theory has been built up. A fundamental connection was noticed between the module theory, and the [[p-adic L-function]]s that were defined in the 1960s by [[Tomio Kubota|Kubota]] and Leopoldt. The latter begin from the [[Bernoulli number]]s, and use [[interpolation]] to define p-adic analogues of the [[Dirichlet L-function]]s. It became clear that the theory had prospects of moving ahead finally from Kummer's century-old results on [[regular prime]]s.
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| Iwasawa formulated the [[main conjecture of Iwasawa theory]] as an assertion that two methods of defining p-adic L-functions (by module theory, by interpolation) should coincide, as far as that was well-defined. This was proved by {{harvtxt|Mazur|Wiles|1984}} for '''Q''', and for all [[totally real number field]]s by {{harvtxt|Wiles|1990}}. These proofs were modeled upon [[Ken Ribet]]'s proof of the converse to Herbrand's theorem (so-called [[Herbrand-Ribet theorem]]).
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| [[Karl Rubin]] found a more elementary proof of the Mazur-Wiles theorem by using Kolyvagin's [[Euler system]]s, described in {{harvtxt|Lang|1990}} and {{harvtxt|Washington|1997}}, and later proved other generalizations of the main conjecture for imaginary quadratic fields.
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| ==Generalizations==
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| The Galois group of the infinite tower, the starting field, and the sort of arithmetic module studied can all be varied. In each case, there is a ''main conjecture'' linking the tower to a ''p''-adic L-function. | |
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| In 2002, Chris Skinner and Eric Urban claimed a proof of a ''main conjecture'' for [[General linear group|GL]](2). In 2010, they posted a preprint {{harv|Skinner|Urban|2010}}.
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| ==See also==
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| *[[Ferrero–Washington theorem]]
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| *[[Tate module of a number field]]
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| == References ==
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| * {{citation | first1=J. | last1=Coates | authorlink1=John Coates (mathematician) | first2=R. | last2=Sujatha | authorlink2=Sujatha Ramdorai | title=Cyclotomic Fields and Zeta Values | series=Springer Monographs in Mathematics | publisher=[[Springer-Verlag]] | year=2006 | isbn=3-540-33068-2 | zbl=1100.11002 }}
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| *{{Citation | last1=Greenberg | first1=Ralph | author1-link=Ralph Greenberg | editor1-last=Miyake | editor1-first=Katsuya | title=Class field theory---its centenary and prospect (Tokyo, 1998) | url=http://www.math.washington.edu/~greenber/iwhi.ps | publisher=Math. Soc. Japan | location=Tokyo | series=Adv. Stud. Pure Math. | isbn=978-4-931469-11-2 | mr=1846466 | year=2001 | volume=30 | chapter=Iwasawa theory---past and present | pages=335–385 | zbl=0998.11054 }}
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| *{{Citation | last1=Iwasawa | first1=Kenkichi | authorlink=Kenkichi Iwasawa | title=On Γ-extensions of algebraic number fields | doi=10.1090/S0002-9904-1959-10317-7 | mr=0124316 | year=1959 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=65 | issue=4 | pages=183–226| zbl=0089.02402 | issn=0002-9904 }}
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| *{{Citation | last1=Kato | first1=Kazuya | author1-link=Kazuya Kato | editor1-last=Sanz-Solé | editor1-first=Marta | editor2-last=Soria | editor2-first=Javier | editor3-last=Varona | editor3-first=Juan Luis | editor4-last=Verdera | editor4-first=Joan | title=International Congress of Mathematicians. Vol. I | url=http://www.icm2006.org/proceedings/Vol_I/18.pdf | publisher=Eur. Math. Soc., Zürich | isbn=978-3-03719-022-7 | doi=10.4171/022-1/14 | mr=2334196 | year=2007 | chapter=Iwasawa theory and generalizations | pages=335–357}}
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| * {{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Cyclotomic fields I and II | url=http://books.google.com/books?isbn=0-387-96671-4 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=Combined 2nd | series=Graduate Texts in Mathematics | isbn=978-0-387-96671-7 | year=1990 | volume=121 | zbl=0704.11038 | others=With an appendix by [[Karl Rubin]] }}
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| *{{Citation | last1=Mazur | first1=Barry | author1-link=Barry Mazur | last2=Wiles | first2=Andrew | author2-link=Andrew Wiles | title=Class fields of abelian extensions of '''Q''' | doi=10.1007/BF01388599 | mr=742853 | year=1984 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=76 | issue=2 | pages=179–330 | zbl=0545.12005 }}
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| *{{Citation
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| | last1=Neukirch | first1=Jürgen | author-link=Jürgen Neukirch
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| | last2=Schmidt | first2=Alexander | last3=Wingberg | first3=Kay
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| | title=Cohomology of Number Fields | chapter=
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| | publisher=[[Springer-Verlag]] | location=Berlin
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| | series=''Grundlehren der Mathematischen Wissenschaften''
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| | volume=323 | year=2008 | page=
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| | isbn=978-3-540-37888-4 | id={{MathSciNet | id = 2392026 }}
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| | zbl= 1136.11001 | edition=Second
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| }}
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| *{{Citation | last1=Rubin | first1=Karl | title=The ‘main conjectures’ of Iwasawa theory for imaginary quadratic fields | doi=10.1007/BF01239508 | year=1991 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=103 | issue=1 | pages=25–68 | unused_data=DUPLICATE DATA: doi=10.1007/BF01239508 | zbl=0737.11030 }}
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| *{{citation| last=Skinner| first=Chris| last2=Urban| first2=Éric| title=The Iwasawa main conjectures for GL<sub>2</sub>| year=2010| url=http://www.math.columbia.edu/%7Eurban/eurp/MC.pdf| page=219}}
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| *{{Citation | last1=Washington | first1=Lawrence C. | title=Introduction to cyclotomic fields | url=http://books.google.com/books?isbn=0-387-94762-0 | publisher=Springer-Verlag | location=Berlin, New York | edition=2nd | series=Graduate Texts in Mathematics | isbn=978-0-387-94762-4 | year=1997 | volume=83}}
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| * {{Citation | author = [[Andrew Wiles]]| year = 1990 | title = The Iwasawa Conjecture for Totally Real Fields | journal = Annals of Mathematics | volume = 131 | issue = 3 | pages = 493–540 | doi = 10.2307/1971468 | publisher = Annals of Mathematics | ref = harv | postscript = . | jstor = 1971468 | zbl=0719.11071 }}
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| ==Further reading==
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| * {{citation | last=de Shalit | first=Ehud | title=Iwasawa theory of elliptic curves with complex multiplication. ''p''-adic ''L'' functions | series=Perspectives in Mathematics | volume=3 | location=Boston etc. | publisher=Academic Press | year=1987 | isbn=0-12-210255-X | zbl=0674.12004 }}
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| ==External links==
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| *{{Springer|title=Iwasawa theory|id=i/i130090}}
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| {{L-functions-footer}}
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| [[Category:Field theory]]
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| [[Category:Cyclotomic fields]]
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| [[Category:Class field theory]]
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This Halloween, make a fresh custom. Instead of it being a day, or days, of indulging inside the favorite sugary treats, make it a day for creating healthy resolutions.
Now regarding calorie intake, the bulky bodybuilder must reduce it. What you need to do first is determine what a basal metabolic rate is (BMR) is. This really is the calorie amount a body burns at rest. There is a fast plus easy-to-use free bmr calculator we can employ on my webpage. Just click on the link under this short article to check it out. Once you learn what the BMR is you need to take in at least 500 nevertheless no over 1000 calories lower than this number. Keeping the calorie expenditure inside this range may aid we burn off the fat without muscle together with it. Plus, you'll keep the metabolic rate significant for maximum fat reduction. This is how to receive ripped rapidly with calorie intake in the event you happe to be bulky.
Food items rich inside fibers: The food items that are especially quite rich in fibers will improve basal metabolic rate at a steady pace. Actually, the fiber content inside the food items would assist to process or break the food particles at a steady pace. Further, you are able to even add protein wealthy items to strengthen the body muscles. Acai berry, prunes plus grapes may additionally be further added in your diet chart.
If you're eating fewer calories than you're burning, you'll lose fat. How much? Well, consider it this way - by burning an extra 500 calories a day, you'll lose one pound every week. Slow progress, sure, however, it's a ideal illustration of how, by keeping track of what you're doing, you can succeed at losing weight.
The daily bmr plus the calories you burn for sport and different escapades make the total amount of calories you burn a day. This really is precisely why BMR is so significant - it helps you plan a weight loss or weight gain program.
Most immediate weight reduction diets don't include any form of bodily exercise. They motivate you to lose weight rapidly by eating pretty little. While which will lead to a rapid initial weight reduction, the fat usually come back when you start to eat usually again.
In summary, we can do a lot to better the metabolism. Lifestyle change plus consistent frequent exercise, all may improve metabolic rate.