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Kronecker's Jugendtraum
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(Definition)
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Kronecker's Jugendtraum (Jugendtraum is German for ``youthful dream'') describes a central problem in class field theory, to explicitly describe the abelian extensions of an arbitrary number field $K$ in terms of values of transcendental functions.
Class field theory gives a complete solution to this problem in the case where $K=\mathbb{Q}$ the field of rational numbers. Specifically, the Kronecker-Weber theorem gives that any abelian number field sits inside one of the cyclotomic fields $\mathbb{Q}(\zeta_n)$ for some $n$ Refining this only slightly gives that we can explicitly generate all abelian extensions of $\mathbb{Q}$ by
adjoining values of the transcendental function $e^{2\pi iz}$ for certain points $z\in \Q/\Z$
A slightly more complicated example is when $K$ is a quadratic imaginary extension of $\mathbb{Q}$ in which case Kronecker's Jugendtraum has been solved by the theory of ``complex multiplication'' (see CM-field). The specific transcendental functions which generate all these abelian extensions are the $j$ function (as in elliptic curves) and Weber's $w$ function.
Though there are partial results in the cases of CM-fields or real quadratic fields, the problem is largely still open, and earned great prestige by being included as Hilbert's twelfth problem.
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"Kronecker's Jugendtraum" is owned by mathcam.
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Cross-references: real quadratic fields, elliptic curves, CM-field, extension, imaginary, points, generate, cyclotomic fields, Kronecker-Weber theorem, rational numbers, solution, transcendental functions, number field, abelian extensions, theory, field, class
This is version 4 of Kronecker's Jugendtraum, born on 2005-02-08, modified 2005-03-18.
Object id is 6726, canonical name is KroneckersJugendtraum.
Accessed 3159 times total.
Classification:
| AMS MSC: | 11R37 (Number theory :: Algebraic number theory: global fields :: Class field theory) |
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Pending Errata and Addenda
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