# Kronecker-Weber theorem

The following theorem classifies the possible http://planetmath.org/node/AbelianExtensionabelian extensions of $\mathbb{Q}$.

###### Theorem 1 (Kronecker-Weber Theorem).

Let $L/\mathbb{Q}$ be a finite http://planetmath.org/node/AbelianExtensionabelian 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 http://planetmath.org/node/AbelianExtensionabelian 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. 1.

$K(j(E))$ is the Hilbert class field of $K$.

2. 2.

If $j(E)\neq 0,1728$ then the maximal http://planetmath.org/node/AbelianExtensionabelian extension of $K$ is given by:

 $K^{ab}=K(j(E),h(E_{\operatorname{torsion}}))$

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:

 $h(P)=\begin{cases}x(P),\text{ if j(E)\neq 0,1728};\\ x^{2}(P),\text{ if j(E)=1728};\\ x^{3}(P),\text{ if j(E)=0}.\end{cases}$

## References

• 1 S. Lang, , Springer-Verlag, New York.
• 2 Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, New York.
 Title Kronecker-Weber theorem Canonical name KroneckerWeberTheorem Date of creation 2013-03-22 13:52:41 Last modified on 2013-03-22 13:52:41 Owner alozano (2414) Last modified by alozano (2414) Numerical id 6 Author alozano (2414) Entry type Theorem Classification msc 11R20 Classification msc 11R37 Classification msc 11R18 Related topic ComplexMultiplication Related topic AbelianExtension Related topic PrimeIdealDecompositionInCyclotomicExtensionsOfMathbbQ Related topic NumberField Related topic CyclotomicExtension Related topic ArithmeticOfEllipticCurves Defines abelian extensions of quadratic imaginary number fields Defines Weber function