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 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 A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

Progress made

The problem was partially solved by Emil Artin (1924; 1927; 1930) by establishing the

Artin reciprocity law The Artin reciprocity law, which was established by Emil Artin in a series of papers (1924; 1927; 1930), is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line ...

which deals with

abelian extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvab ...

s of

s. Together with the work of

Teiji Takagi Teiji Takagi (高木貞治 ''Takagi Teiji'', April 21, 1875 – February 28, 1960) was a Japanese mathematician, best known for proving the Takagi existence theorem in class field theory. The Blancmange curve, the graph of a nowhere-differentiab ...

and Helmut Hasse (who established the more general Hasse reciprocity law), this led to the development of the

class field theory In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credit ...

, 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 Kronecker's Jugendtraum or Hilbert's twelfth problem, of the 23 mathematical Hilbert problems, is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any base number field. That is, it asks for analog ...

, is one of the long-standing challenges in number theory and is far from being complete.

References

External links

English translation of Hilbert's original address
{{Hilbert's problems Algebraic number theory Unsolved problems in number theory #09

Progress made

See also

References

External links