Description: This problem asks to extend the Kronecker-Weber Theorem on abelian extensions of the rational numbers to any base number field. The current Kronecker-Weber Theorem does this for the case of any imaginary quadratic field, showing that every algebraic integer whose Galois group is abelian can be expressed as a sum of roots of unity with rational coefficients.
Source: Wikipedia
Status: The work done by Shimura and Taniyama in the complex multiplication of abelian varieties gave rise to abelian extensions of CM-fields. Additionally, Stark’s conjecture resulted in a great conjectural development of L-functions, and is capable of producing concrete, numerical results. However, this problem still remains unresolved.
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