# Hilbert's ninth problem

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{{Short description|On the reciprocity law in algebraic number fields}}
{{No footnotes|date=January 2021}}
'''Hilbert's ninth problem''', from the list of 23 [Hilbert's problems](/source/Hilbert's_problems) (1900), asked to find the most general [reciprocity law](/source/reciprocity_law_(mathematics)) for the [norm residues](/source/Hilbert_symbol) of ''k''-th order in a general [algebraic number field](/source/algebraic_number_field), where ''k'' is a power of a [prime](/source/prime_number).

The problem was partially solved for abelian extensions by [Artin reciprocity](/source/Artin_reciprocity) and class field theory for [abelian extensions](/source/abelian_extensions) of number fields. Generalization of these results to [non-abelian class field theory](/source/non-abelian_class_field_theory) seems to be one of the most challenging problems in [algebraic number theory](/source/algebraic_number_theory), which is also related with [Hilbert's twelfth problem](/source/Hilbert's_twelfth_problem).

== Progress made ==

The problem was partially solved by {{harvtxt|Artin|1924}}, {{harvtxt|Artin|1927}} and {{harvtxt|Artin|1930}} by establishing the Artin reciprocity law which deals with abelian extensions of number fields. Together with the work of [Teiji Takagi](/source/Teiji_Takagi) and [Helmut Hasse](/source/Helmut_Hasse) (who established the more general Hasse reciprocity law), this led to the development of the [class field theory](/source/class_field_theory), realizing Hilbert's program in an abstract fashion. Certain explicit formulas for norm residues were later found by [Igor Shafarevich](/source/Igor_Shafarevich) (1948; 1949; 1950).

[Robert Langlands](/source/Robert_Langlands) in his 1967 letter to [Andre Weil](/source/Andre_Weil) made conjecture about nonabelian reciprocity involving [Artin L-function](/source/Artin_L-function)s and [automorphic L-function](/source/automorphic_L-function)s: for finite number field extension <math>L/K</math>, let <math>\rho</math> be irreducible representation  of [Galois group](/source/Galois_group) of this extension and <math>\mathbf{A}_K</math> be [adele ring](/source/adele_ring) of <math>K</math>. If <math>L(s ,\rho , L/K)</math> is Artin L-function for that Galois group and this representation, then Langlands reciprocity conjecture says that there exists [automorphic cuspidal representation](/source/automorphic_form) <math>\pi</math> of [general linear group](/source/general_linear_group) <math>GL(n,A_K)</math> that:
:<math> L(s, \pi) = L(s ,\rho , L/K) </math>
where <math>L(s, \pi)</math> is automorphic L-function for this representation. This conjecture generalizes Artin reciprocity and became starting point for much more general [Langlands program](/source/Langlands_program). Despite some results towards Langlands program, this conjecture seems to be far from proven, but stands as the best proposition of solution for Hilbert's ninth problem.

==See also==
*[List of unsolved problems in mathematics](/source/List_of_unsolved_problems_in_mathematics)

==References==
*{{cite journal
 | last = Artin | first = Emil | author-link = Emil Artin
 | year = 1924
 | title = Über eine neue Art von L-Reihen
 | journal = [Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg](/source/Abhandlungen_aus_dem_Mathematischen_Seminar_der_Universit%C3%A4t_Hamburg)
 | volume = 3
 | pages = 89–108
}}
*{{cite journal
 | last = Artin | first = Emil | author-link = Emil Artin
 | year = 1927
 | title = Beweis des allgemeinen Reziprozitätsgesetzes
 | journal = Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
 | volume = 5
 | pages = 353–363}}
*{{cite journal
 | last = Artin | first = Emil | author-link = Emil Artin
 | year = 1930
 | title = Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetzes
 | journal = Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
 | volume = 7
 | pages = 46–51}}
* {{cite book | editor=Felix E. Browder | editor-link=Felix Browder | title=Mathematical Developments Arising from Hilbert Problems | series=[Proceedings of Symposia in Pure Mathematics](/source/Proceedings_of_Symposia_in_Pure_Mathematics) | volume=XXVIII.2 | year=1976 | publisher=[American Mathematical Society](/source/American_Mathematical_Society) | isbn=0-8218-1428-1 | first=John | last=Tate | authorlink=John Tate (mathematician) | chapter=Problem 9: The general reciprocity law | pages=311–322 }}

== External links ==
* [http://aleph0.clarku.edu/~djoyce/hilbert/problems.html#prob9 English translation of Hilbert's original address]

{{Hilbert's problems}}

Category:Algebraic number theory
Category:Unsolved problems in number theory
#09

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Adapted from the Wikipedia article [Hilbert's ninth problem](https://en.wikipedia.org/wiki/Hilbert's_ninth_problem) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Hilbert's_ninth_problem?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.