Hilbert's ninth problem
Encyclopedia
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
.
(1924; 1927; 1930) by establishing the Artin reciprocity law which deals with abelian extension
s of algebraic number field
s. 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.
Hilbert's problems
Hilbert's problems form a list of twenty-three problems in mathematics published by German mathematician David Hilbert in 1900. The problems were all unsolved at the time, and several of them were very influential for 20th century mathematics...
(1900), asked to find the most general reciprocity law for the norm residues
Hilbert symbol
In mathematics, given a local field K, such as the fields of reals or p-adic numbers, whose multiplicative group of non-zero elements is K×, the Hilbert symbol is an algebraic construction, extracted from reciprocity laws, and important in the formulation of local class field theory...
of k-th order in a general algebraic number field
Algebraic number field
In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...
, where k is a power of a prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
.
Progress made
The problem was partially solved by Emil ArtinEmil Artin
Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...
(1924; 1927; 1930) by establishing the Artin reciprocity law which deals with abelian extension
Abelian extension
In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is a cyclic group, we have a cyclic extension. More generally, a Galois extension is called solvable if its Galois group is solvable....
s of algebraic number field
Algebraic number field
In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...
s. Together with the work of Teiji Takagi
Teiji Takagi
Teiji Takagi was a Japanese mathematician, best known for proving the Takagi existence theorem in class field theory....
and Helmut Hasse
Helmut Hasse
Helmut Hasse 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 classfield theory and diophantine geometry , and to local zeta functions.-Life:He was born in Kassel, and died in...
(who established the more general Hasse reciprocity law), this led to the development of the class field theory
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
, realizing Hilbert's program in an abstract fashion. Certain explicit formulas for norm residues were later found by Igor Shafarevich
Igor Shafarevich
Igor Rostislavovich Shafarevich is a Soviet and Russian mathematician, founder of a school of algebraic number theory and algebraic geometry in the USSR, and a political writer. He was also an important dissident figure under the Soviet regime, a public supporter of Andrei Sakharov's Human Rights...
(1948; 1949; 1950).
The non-abelian generalization
Non-abelian class field theory
In mathematics, non-abelian class field theory is a catchphrase, meaning the extension of the results of class field theory, the relatively complete and classical set of results on abelian extensions of any number field K, to the general Galois extension L/K...
, also connected with Hilbert's twelfth problem
Hilbert's twelfth problem
Kronecker's Jugendtraum or Hilbert's twelfth problem, of the 23 mathematical Hilbert problems, is the extension of Kronecker–Weber theorem on abelian extensions of the rational numbers, to any base number field...
, is one of the long-standing challenges in number theory and is far from being complete.