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[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.



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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

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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 7412 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)
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