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Iwasawa worked with so-called-extensions: infinite extensions of a number field with Galois group isomorphic to the additive group of p-adic integers for some prime p. Every closed subgroup of is of the form, so by Galois theory, a-extension is the same thing as a tower of fields such that.

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## Some Related Sentences

Iwasawa and worked

__Iwasawa__spent

**the**next two years at Institute

**for**Advanced Study in Princeton

**,**and in Spring

**of**1952 was offered

**a**job at

**the**Massachusetts Institute

**of**Technology

**,**where he

__worked__until 1967

**.**

Before

**that**he__worked__on Lie groups and Lie algebras**,**introducing**the**general__Iwasawa__decomposition**.**

Iwasawa and with

Also

**,**starting__with__any compact real**form****of****a**semisimple Lie algebra g its complexification**as****a**real Lie algebra**of**twice**the**dimension splits into g and**a**certain solvable Lie algebra (**the**__Iwasawa__decomposition ), and this provides**a**canonical bicrossproduct quantum**group**associated**to**g**.**For su ( 2 ) one obtains**a**quantum**group**deformation**of****the**Euclidean**group**E ( 3 )**of**motions in 3 dimensions**.**
More generally

**,**__Iwasawa__**theory**asks questions about**the**structure**of****Galois**modules over**extensions**__with__**Galois****group****a****p-adic**Lie**group****.**
Combining this

__with__**the**p-parity theorem**by**and**the**announced proof**of****the**main conjecture**of**__Iwasawa__**theory****for**GL ( 2 )**by****,**they conclude**that****a**positive proportion**of**elliptic curves over Q have analytic rank zero**,**and hence**,****by****,**satisfy**the**Birch and Swinnerton-Dyer conjecture**.**
For

**a**general semisimple Lie**group****,****the**decomposition**is****the**__Iwasawa__decomposition**of**G**as**G = KAN in which K occurs in**a**product__with__**a**contractible**subgroup**AN**.**
However

**,**this**same**year__Iwasawa__became sick__with__pleurisy**,**and was unable**to**return**to**his position at**the**university until April 1947**.**

Iwasawa and infinite

In

**the**last ten years he has focused on**the**study**of**various aspects**of**non-commutative__Iwasawa__**theory****,****for**instance**,****the**study**of****the**arithmetic**of**elliptic curves in nonabelian__infinite__**extensions****.**
In

**number****theory****,**__Iwasawa__**theory****is****the**study**of**objects**of**arithmetic interest over__infinite__towers**of****number****fields****.**

Iwasawa and number

Tate's thesis ( 1950 ) on Fourier analysis in

__number__**fields**has become one**of****the**ingredients**for****the**modern**theory****of**automorphic forms and their L-functions**,**notably**by**its use**of****the**adele ring**,**its self-duality and harmonic analysis on it ; independently and**a**little earlier**,**Kenkichi__Iwasawa__obtained**a**similar**theory****.**
Kenkichi

__Iwasawa__(__Iwasawa__Kenkichi**,**September 11**,**1917 – October 26**,**1998 ) was**a**Japanese mathematician who**is**known**for**his influence on algebraic__number__**theory****.**

Iwasawa and field

Iwasawa and Galois

* Mathematics

**:**Approximation**theory**— Arakelov**theory**— Asymptotic**theory**— Bifurcation**theory**— Catastrophe**theory**— Category**theory**— Chaos**theory**— Choquet**theory**— Coding**theory**— Combinatorial game**theory**— Computability**theory**— Computational complexity**theory**— Deformation**theory**— Dimension**theory**— Ergodic**theory**— Field**theory**—__Galois__**theory**— Game**theory**— Graph**theory**— Group**theory**— Hodge**theory**— Homology**theory**— Homotopy**theory**— Ideal**theory**— Intersection**theory**— Invariant**theory**—__Iwasawa__**theory**— K-theory — KK-theory — Knot**theory**— L-theory — Lie**theory**— Littlewood – Paley**theory**— Matrix**theory**— Measure**theory**— Model**theory**— Morse**theory**— Nevanlinna**theory**— Number**theory**— Obstruction**theory**— Operator**theory**— PCF**theory**— Perturbation**theory**— Potential**theory**— Probability**theory**— Ramsey**theory**— Rational choice**theory**— Representation**theory**— Ring**theory**— Set**theory**— Shape**theory**— Small cancellation**theory**— Spectral**theory**— Stability**theory**— Stable**theory**— Sturm – Liouville**theory**— Twistor**theory**
It began

**as****a**__Galois__module**theory****of**ideal class groups**,**initiated**by**Kenkichi__Iwasawa__**,**in**the**1950s**,****as**part**of****the****theory****of**cyclotomic**fields****.**__Iwasawa__studied classical

__Galois__modules over

**by**asking questions about

**the**structure

**of**modules over

**.**

In order

**to**get an interesting__Galois__module here**,**__Iwasawa__took**the**ideal class**group****of****,**and let be its p-torsion part**.**
Ralph Greenberg has generalized

**the**notion**of**Selmer**group****to**more general**p-adic**__Galois__representations and**to****p-adic**variations**of**motives in**the**context**of**__Iwasawa__**theory****.**

Iwasawa and group

* The

__Iwasawa__decomposition G = KAN**of****a**semisimple__group__G**as****the**product**of**compact**,**abelian**,**and nilpotent subgroups generalises**the**way**a**square real matrix can be written**as****a**product**of**an orthogonal matrix and an upper triangular matrix (**a**consequence**of**Gram – Schmidt orthogonalization ).
In mathematics

**,****the**__Iwasawa__decomposition KAN**of****a**semisimple Lie__group__generalises**the**way**a**square real matrix can be written**as****a**product**of**an orthogonal matrix and an upper triangular matrix (**a**consequence**of**Gram-Schmidt orthogonalization ).

Iwasawa and p-adic

The main conjecture

**of**__Iwasawa__**theory**was formulated**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**.**

Iwasawa and for

These elements — named Beilinson elements after Alexander Beilinson who introduced them in — were used

**by**Kazuya Kato in**to**prove one divisibility in Barry Mazur's main conjecture**of**__Iwasawa__**theory**__for__elliptic curves**.**__Iwasawa__decompositions also hold

__for__

**some**disconnected semisimple groups G

**,**where K becomes

**a**( disconnected ) maximal compact

**subgroup**provided

**the**center

**of**G

**is**finite

**.**

__Iwasawa__

**is**perhaps best known

__for__introducing what

**is**now called

__Iwasawa__

**theory**

**,**which developed from researches on cyclotomic

**fields**from

**the**later parts

**of**

**the**1950s

**.**

Iwasawa and some

* In Japan

**,**according**to**M**.**__Iwasawa__at**the**National Institute**of**Population and Social Security Research**,**less than 3 %**of**females between 25-29 are currently cohabiting**,**but more than 1 in 5 have had__some__experience**of**an unmarried partnership**,**including cohabitation**.**

Iwasawa and p

Ribet's methods were pushed further

**by**Barry Mazur and Andrew Wiles in order**to**prove**the**Main Conjecture**of**__Iwasawa__**theory**,< ref >**a**corollary**of**which**is****a**strengthening**of****the**Herbrand-Ribet theorem**:****the**power**of**__p__dividing B < sub >__p__− n </ sub >**is**exactly**the**power**of**__p__dividing**the**order**of**G < sub > n </ sub >.

Iwasawa and .

A more recent

__Iwasawa__study has shown**that**there has been**a**recent emergence**of**non-marital cohabitation__.__
Via

**the****theory****of**zeta integrals initiated**by**Kenkichi__Iwasawa__and**by**John Tate in Tate's thesis it**is**related**to****the**study**of****the**zeta function**of**global**fields**__.__
* Borel

**,**A .; Chowla**,**S .; Herz**,**C__.__S .;__Iwasawa__**,**K .; Serre**,**J .- P__.__Seminar on complex multiplication__.__

Iwasawa and is

In 1950

**,**__Iwasawa__was invited**to**Cambridge**,**Massachusetts**to**give**a**lecture at**the**International Congress**of**Mathematicians on his method**to**study Dedekind zeta functions using integration over ideles and duality**of**adeles ; this method was also independently obtained**by**John Tate and it__is__sometimes called Tate's thesis or**the**Iwasawa-Tate**theory****.**0.160 seconds.