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.

Progress made

The problem was partially solved by Emil Artin by establishing the Artin reciprocity law which deals with abelian extensions of algebraic number fields.^[1]^[2]^[3] Together with the work of Teiji Takagi and Helmut Hasse (who established the more general Hasse reciprocity law), this led to the development of the class field theory, realizing Hilbert's program in an abstract fashion. Certain explicit formulas for norm residues were later found by Igor Shafarevich (1948; 1949; 1950).

The non-abelian generalization, also connected with Hilbert's twelfth problem, is one of the long-standing challenges in number theory and is far from being complete.

^ Artin, Emil (1924). "Über eine neue Art von L-Reihen". Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. 3: 89–108.
^ Artin, Emil (1927). "Beweis des allgemeinen Reziprozitätsgesetzes". Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. 5: 353–363.
^ Artin, Emil (1930). "Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetzes". Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. 7: 46–51.

Tate, John (1976). "Problem 9: The general reciprocity law". In Felix E. Browder (ed.). Mathematical Developments Arising from Hilbert Problems. Proceedings of Symposia in Pure Mathematics. Vol. XXVIII.2. American Mathematical Society. pp. 311–322. ISBN 0-8218-1428-1.