PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (198)
Orphanage
Unclass'd
Unproven (490)
Corrections (7)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
[parent] Kronecker-Weber theorem (Theorem)

The following theorem classifies the possible abelian extensions of $ \mathbb{Q}$.

Theorem 1 (Kronecker-Weber Theorem)   Let $ L/\mathbb{Q}$ be a finite abelian extension, then $ L$ is contained in a cyclotomic extension, i.e. there is a root of unity $ \zeta$ such that $ L \subseteq \mathbb{Q}(\zeta)$.

In a similar fashion to this result, the theory of elliptic curves with complex multiplication provides a classification of abelian extensions of quadratic imaginary number fields:

Theorem 2   Let $ K$ be a quadratic imaginary number field with ring of integers $ \mathcal{O}_K$. Let $ E$ be an elliptic curve with complex multiplication by $ \mathcal{O}_K$ and let $ j(E)$ be the $ j$-invariant of $ E$. Then:
  1. $ K(j(E))$ is the Hilbert class field of $ K$.
  2. If $ j(E)\neq 0,1728$ then the maximal abelian extension of $ K$ is given by:
    $\displaystyle K^{ab}=K(j(E),h(E_{\operatorname{torsion}}))$
    where $ h(E_{\operatorname{torsion}})$ is the set of $ x$-coordinates of all the torsion points of $ E$.

Note: The map $ h\colon E \to \mathbb{C}$ is called a Weber function for $ E$. We can define a Weber function for the cases $ j(E)=0,1728$ so the theorem holds true for those two cases as well. Assume $ E\colon y^2=x^3+Ax+B$, then:

\begin{displaymath}h(P)= \begin{cases} x(P) ,\text{ if $j(E)\neq 0, 1728$};\ x... ...t{ if $j(E)=1728$};\ x^3(P) ,\text{ if $j(E)=0$}. \end{cases}\end{displaymath}

Bibliography

1
S. Lang, Algebraic Number Theory, Springer-Verlag, New York.
2
Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, New York.



"Kronecker-Weber theorem" is owned by alozano.
(view preamble)

View style:

See Also: complex multiplication, abelian extension, prime ideal decomposition in cyclotomic extensions of $\mathbb{Q}$, number field, cyclotomic extension, the arithmetic of elliptic curves

Also defines:  abelian extensions of quadratic imaginary number fields, Weber function

This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: map, points, torsion, Hilbert class field, ring of integers, quadratic imaginary number fields, complex multiplication, elliptic curves, theory, similar, root of unity, cyclotomic extension, contained, finite
There are 7 references to this entry.

This is version 3 of Kronecker-Weber theorem, born on 2003-08-19, modified 2006-10-02.
Object id is 4620, canonical name is KroneckerWeberTheorem.
Accessed 7425 times total.

Classification:
AMS MSC: 11R18 (Number theory :: Algebraic number theory: global fields :: Cyclotomic extensions)
 11R37 (Number theory :: Algebraic number theory: global fields :: Class field theory)
 11R20 (Number theory :: Algebraic number theory: global fields :: Other abelian and metabelian extensions)
Pending Errata and Addenda
None.
[ View all 2 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | prove | add result | add corollary | add example | add (any)