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Kronecker-Weber theorem
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(Theorem)
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The following theorem classifies the possible abelian extensions of $\Rats$ .
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:
- $K(j(E))$ is the Hilbert class field of $K$ .
- If $j(E)\neq 0,1728$ then the maximal abelian 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 $h\colon E \to \Complex$ 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:
- 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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"Kronecker-Weber theorem" is owned by alozano.
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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, theorem
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 9255 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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Pending Errata and Addenda
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