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 by Emil Artin (1924; 1927; 1930) by establishing the Artin reciprocity law which deals with abelian extensions of algebraic number fields. 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).
- 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. XXVIII.2. American Mathematical Society. pp. 311–322. ISBN 0-8218-1428-1.