Hilbert's ninth problem

Hilbert's ninth problem, from the list of 23 Hilbert's problems (1900), asked to find the most general reciprocity law for the norm residues of k-th order in a general algebraic number field, where k is a power of a prime.

The problem was partially solved for abelian extensions by Artin reciprocity and class field theory for abelian extensions of number fields. Generalization of these results to non-abelian class field theory seems to be one of the most challenging problems in algebraic number theory, which is also related with Hilbert's twelfth problem.

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