Hilbert's ninth problem, from the list of 23
Hilbert's problems
Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the pro ...
(1900), asked to find the most general
reciprocity law for the
norm residues of ''k''-th order in a general
algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
, where ''k'' is a power of a
prime.
Progress made
The problem was partially solved by
Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent.
Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing lar ...
(1924; 1927; 1930) by establishing the
Artin reciprocity law which deals with
abelian extensions of
algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
s. Together with the work of
Teiji Takagi and
Helmut Hasse
Helmut Hasse (; 25 August 1898 – 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of ''p''-adic numbers to local class field theory and ...
(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
Igor Rostislavovich Shafarevich (russian: И́горь Ростисла́вович Шафаре́вич; 3 June 1923 – 19 February 2017) was a Soviet and Russian mathematician who contributed to algebraic number theory and algebraic geometry. ...
(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.
See also
*
List of unsolved problems in mathematics
References
*
External links
English translation of Hilbert's original address
{{Hilbert's problems
Algebraic number theory
Unsolved problems in number theory
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