Editing Hilbert's problems (section)

==Table of problems==
Hilbert's 23 problems are (for details on the solutions and references, see the articles that are linked to in the first column):
{| class="wikitable sortable"
|-
! Problem
!class="unsortable"| Brief explanation
! Status
! Year solved
|-
|style="text-align:center"| [[Hilbert's first problem|1st]]
| The [[continuum hypothesis]] (that is, there is no [[Set (mathematics)|set]] whose [[cardinality]] is strictly between that of the [[integer]]s and that of the [[real number]]s)
|{{partial|align=left|{{sort|2|}} Proven to be impossible to prove or disprove within [[Zermelo–Fraenkel set theory]] with or without the [[axiom of choice]] (provided Zermelo–Fraenkel set theory is [[consistency|consistent]], i.e., it does not contain a contradiction). There is no consensus on whether this is a solution to the problem.}}
|style="text-align:center"| 1940, 1963
|-
|style="text-align:center"| [[Hilbert's second problem|2nd]]
| Prove that the [[axiom]]s of [[arithmetic]] are [[consistency|consistent]].
|{{partial|align=left|{{sort|2|}} There is no consensus on whether results of [[Kurt Gödel|Gödel]] and [[Gerhard Gentzen|Gentzen]] give a solution to the problem as stated by Hilbert. Gödel's [[Gödel's incompleteness theorems|second incompleteness theorem]], proved in 1931, shows that no proof of its consistency can be carried out within arithmetic itself. Gentzen proved in 1936 that the consistency of arithmetic follows from the [[well-founded relation|well-foundedness]] of the [[epsilon numbers (mathematics)|ordinal&nbsp;''ε''<sub>0</sub>]].}}
|style="text-align:center"| 1931, 1936
|-
|style="text-align:center"| [[Hilbert's third problem|3rd]]
| Given any two [[polyhedron|polyhedra]] of equal volume, is it always possible to cut the first into finitely many polyhedral pieces that can be reassembled to yield the second?
|{{yes|align=left|{{sort|1|}} Resolved. Result: No, proved using [[Dehn invariant]]s.}}
|style="text-align:center"| 1900
|-
|style="text-align:center"| [[Hilbert's fourth problem|4th]]
| Construct all [[metric space|metrics]] where lines are [[geodesic]]s.
|{{unknown|align=left|style=font-size:inherit|{{sort|4|}} Too vague to be stated resolved or not.{{refn|According to Gray, most of the problems have been solved. Some were not defined completely, but enough progress has been made to consider them "solved"; Gray lists the fourth problem as too vague to say whether it has been solved.|group=lower-alpha}} }}
|style="text-align:center"| —
|-
|style="text-align:center"| [[Hilbert's fifth problem|5th]]
| Are continuous [[group (mathematics)|groups]] automatically [[Lie group|differential groups]]?
|{{partial|align=left|{{sort|2|}} Resolved by [[Andrew Gleason]], assuming one interpretation of the original statement. If, however, it is understood as an equivalent of the [[Hilbert–Smith conjecture]], it is still unsolved.}}
|style="text-align:center"| 1953?
|-
|style="text-align:center"| [[Hilbert's sixth problem|6th]]
| Mathematical treatment of the [[axiom]]s of [[physics]]:

(a) axiomatic treatment of probability with limit theorems for foundation of [[statistical physics]]

(b) the rigorous theory of limiting processes "which lead from the atomistic view to the laws of motion of continua"
|{{partial|align=left|{{sort|2|}} Unresolved, or partially resolved, depending on how the original statement is interpreted.<ref>{{cite journal |last1=Corry |first1=L. |year=1997 |title=David Hilbert and the axiomatization of physics (1894–1905) |journal=Arch. Hist. Exact Sci. |volume=51 |issue=2 |pages=83–198 |doi=10.1007/BF00375141 |s2cid=122709777 }}</ref> Items&nbsp;(a) and (b) were two specific problems given by Hilbert in a later explanation.<ref name=Hilbert_1902/> [[probability axioms|Kolmogorov's axiomatics]] (1933) is now accepted as standard for the foundations of probability theory. There is some success on the way from the "atomistic view to the laws of motion of continua",<ref>{{cite journal |last1=Gorban |first1=A. N. |author-link=Alexander Nikolaevich Gorban |last2=Karlin |first2=I. |year=2014 |title=Hilbert's 6th Problem: Exact and approximate hydrodynamic manifolds for kinetic equations |journal=Bulletin of the American Mathematical Society |volume=51 |issue=2 |pages=186–246 |arxiv=1310.0406 |doi=10.1090/S0273-0979-2013-01439-3| doi-access= free}}</ref>}} but the transition from classical to quantum physics means that there would have to be two axiomatic formulations, with a clear link between them. [[John von Neumann]] made an early attempt to place [[quantum mechanics]] on a rigorous mathematical basis in his book ''[[Mathematical Foundations of Quantum Mechanics]]'',<ref>{{Cite book |last=Von Neumann |first=John |author-link=John von Neumann |url=https://press.princeton.edu/titles/11352.html |title=Mathematical foundations of quantum mechanics |date=2018 |publisher=Princeton University Press |isbn=978-0-691-17856-1 |editor-last=Wheeler |editor-first=Nicholas A. |location=Princeton Oxford |translator-last=Beyer |translator-first=Robert T.}}</ref> but subsequent developments have occurred, further challenging the axiomatic foundations of quantum physics. 
|style="text-align:center"| 1933–2002?
|-
|style="text-align:center"| [[Hilbert's seventh problem|7th]]
| Is ''a<sup>b</sup>'' [[transcendental number|transcendental]], for [[algebraic number|algebraic]] ''a'' ≠ 0,1 and [[irrational number|irrational]] algebraic ''b'' ?
|{{yes|align=left|{{sort|1|}} Resolved. Result: Yes, illustrated by the [[Gelfond–Schneider theorem]].}}
|style="text-align:center"| 1934
|-
|style="text-align:center"| [[Hilbert's eighth problem|8th]]
| The [[Riemann hypothesis]] ("the real part of any non-[[Triviality (mathematics)|trivial]] [[Zero of a function|zero]] of the [[Riemann zeta function]] is 1/2") and other prime-number problems, among them [[Goldbach's conjecture]] and the [[twin prime conjecture]]
|{{no|align=left|{{sort|3|}} Unresolved.}}
|style="text-align:center"| —
|-
|style="text-align:center"| [[Hilbert's ninth problem|9th]]
| Find the most general law of the [[Quadratic reciprocity|reciprocity theorem]] in any [[algebra]]ic [[number field]].
|{{partial|align=left|{{sort|2|}} Partially resolved. Solved by [[Emil Artin]] in 1927 for [[abelian extension]]s of the [[rational number]]s during the development of [[class field theory]]. The non-abelian case remains unsolved, if one interprets that as meaning [[non-abelian class field theory]].|group=lower-alpha}}
|style="text-align:center"| —
|-
|style="text-align:center"| [[Hilbert's tenth problem|10th]]
| Find an algorithm to determine whether a given polynomial [[Diophantine equation]] with integer coefficients has an integer solution.
|{{yes|align=left|{{sort|1|}} Resolved. Result: Impossible; [[Matiyasevich's theorem]] implies that there is no such algorithm.}}
|style="text-align:center"| 1970
|-
|style="text-align:center"| [[Hilbert's eleventh problem|11th]]
| Solving [[quadratic form]]s with algebraic numerical [[coefficient]]s.
|{{partial|align=left|{{sort|2|}} Partially resolved.<ref name="Hazewinkel">{{cite book |first=Michiel |last=Hazewinkel |date=2009 |title=Handbook of Algebra |publisher=Elsevier |page=69 |isbn=978-0080932811 |volume=6}}</ref>}}
|style="text-align:center"| —
|-
|style="text-align:center"| [[Hilbert's twelfth problem|12th]]
| Extend the [[Kronecker–Weber theorem]] on abelian extensions of the [[rational number]]s to any base number field.
|{{partial|align=left|{{sort|2|}} Partially resolved.<ref>{{cite web |url=https://www.quantamagazine.org/mathematicians-find-polynomial-building-blocks-hilbert-sought-20210525/ |first=Kelsey |last=Houston-Edwards |title=Mathematicians Find Long-Sought Building Blocks for Special Polynomials |date=25 May 2021 }}</ref>}}
|style="text-align:center"| —
|-
|style="text-align:center"| [[Hilbert's thirteenth problem|13th]]
| Solve [[septic equation|7th-degree equation]] using algebraic (variant: continuous) [[mathematical function|functions]] of two [[parameter]]s.
|{{no|align=left|{{sort|3|}} Unresolved. The continuous variant of this problem was solved by [[Vladimir Arnold]] in 1957 based on work by [[Andrei Kolmogorov]] (see [[Kolmogorov–Arnold representation theorem]]), but the algebraic variant is unresolved.{{refn|1=It is not difficult to show that the problem has a partial solution within the space of single-valued analytic functions (Raudenbush). Some authors argue that Hilbert intended for a solution within the space of (multi-valued) algebraic functions, thus continuing his own work on algebraic functions and being a question about a possible extension of the [[Galois theory]] (see, for example, Abhyankar<ref name="Abyankar">{{cite book |url=http://www.emis.de/journals/SC/1997/2/pdf/smf_sem-cong_2_1-11.pdf |first=Shreeram S. |last=Abhyankar |title=Hilbert's Thirteenth Problem |series=Séminaires et Congrès |volume=2 |publisher=Société Mathématique de France |date=1997}}</ref> Vitushkin,<ref>{{cite journal |last1=Vitushkin |first1=Anatoliy G. |title=On Hilbert's thirteenth problem and related questions |journal=Russian Mathematical Surveys |date=2004 |volume=59 |issue=1 |pages=11–25 |doi=10.1070/RM2004v059n01ABEH000698 |publisher=Russian Academy of Sciences|bibcode=2004RuMaS..59...11V |s2cid=250837749 }}</ref> Chebotarev,<ref>{{cite journal |last1=Morozov |first1=Vladimir V. |title=О некоторых вопросах проблемы резольвент |journal=Proceedings of Kazan University |date=1954 |volume=114 |issue=2 |pages=173–187 |url=http://www.mathnet.ru/php/getFT.phtml?jrnid=uzku&paperid=406&what=fullt&option_lang=eng |publisher=Kazan University |language=ru |trans-title=On certain questions of the problem of resolvents}}</ref> and others). It appears from one of Hilbert's papers<ref>{{cite journal |first=David |last=Hilbert |title=Über die Gleichung neunten Grades |journal=Math. Ann. |volume=97 |year=1927 |pages=243–250 |doi=10.1007/BF01447867 |s2cid=179178089 }}</ref> that this was his original intention for the problem.
The language of Hilbert there is "{{lang|de|Existenz von ''algebraischen'' Funktionen|italic=unset}}" ("existence of ''algebraic'' functions").
As such, the problem is still unresolved.|group=lower-alpha}} }}
|style="text-align:center"| —
|-
|style="text-align:center"| [[Hilbert's fourteenth problem|14th]]
| Is the [[invariant theory|ring of invariants]] of an [[algebraic group]] acting on a [[polynomial ring]] always [[Finitely generated algebra|finitely generated]]?
|{{yes|align=left|{{sort|1|}} Resolved. Result: No, a counterexample was constructed by [[Masayoshi Nagata]].}}
|style="text-align:center"| 1959
|-
|style="text-align:center"| [[Hilbert's fifteenth problem|15th]]
| Rigorous foundation of [[Schubert's enumerative calculus]].
|{{partial|align=left|{{sort|2|}} Partially resolved.<ref name="KL">{{cite journal | last1=Kleiman | first1=S.L.| last2=Laksov | first2=Dan |author1-link=Steven Kleiman |author2-link=Dan Laksov|title= Schubert Calculus | publisher=American Mathematical Society|journal = American Mathematical Monthly | volume=79| issue=10 | year=1972 | issn=0377-9017 | doi=10.1080/00029890.1972.11993188 | pages=1061–1082 }}</ref> Haibao Duan and Xuezhi Zhao claimed that this problem is actually resolved.}}
|style="text-align:center"| —
|-
|style="text-align:center"| [[Hilbert's sixteenth problem|16th]]
| Describe relative positions of ovals originating from a [[real number|real]] [[algebraic curve]] and as [[limit cycle]]s of a polynomial [[vector field]] on the plane.
|{{no|align=left|{{sort|3|}} Unresolved, even for algebraic curves of degree&nbsp;8.}}
|style="text-align:center"| —
|-
|style="text-align:center"| [[Hilbert's seventeenth problem|17th]]
| Express a nonnegative [[rational function]] as [[quotient]] of sums of [[Square (algebra)|squares]].
|{{yes|align=left|{{sort|1|}} Resolved. Result: Yes, due to [[Emil Artin]]. Moreover, an upper limit was established for the number of square terms necessary.}}
|style="text-align:center"| 1927
|-
|rowspan=3 style="text-align:center"| [[Hilbert's eighteenth problem|18th]]
| (a) Are there only finitely many essentially different [[space group]]s in ''n''-dimensional Euclidean space?
|{{yes|align=left|{{sort|1|}} Resolved. Result: Yes (by [[Ludwig Bieberbach]])}}
|style="text-align:center"| 1910
|-
| (b) Is there a polyhedron that admits only an [[anisohedral tiling]] in three dimensions?
|{{yes|align=left|{{sort|1|}} Resolved. Result: Yes (by [[Karl Reinhardt (mathematician)|Karl Reinhardt]]).}}
|style="text-align:center"| 1928
|-
| (c) What is the densest [[sphere packing]]?
|{{yes|align=left|{{sort|1|}} Widely believed to be resolved, by [[computer-assisted proof]] (by [[Thomas Callister Hales]]). Result: Highest density achieved by [[Close-packing of equal spheres|close packings]], each with density approximately 74%, such as face-centered cubic close packing and hexagonal close packing.{{refn|Gray also lists the 18th problem as "open" in his 2000 book, because the sphere-packing problem (also known as the [[Kepler conjecture]]) was unsolved, but a solution to it has now been claimed.|group=lower-alpha}}}}
|style="text-align:center"| 1998
|-
|style="text-align:center"| [[Hilbert's nineteenth problem|19th]]
| Are the solutions of regular problems in the [[calculus of variations]] always necessarily [[Analytic function|analytic]]?
|{{yes|align=left|{{sort|1|}} Resolved. Result: Yes, proven by [[Ennio De Giorgi]] and, independently and using different methods, by [[John Forbes Nash]].}}
|style="text-align:center"| 1957
|-
|style="text-align:center"| [[Hilbert's twentieth problem|20th]]
| Do all [[calculus of variations|variational problems]] with certain [[boundary condition]]s have solutions?
|{{partial|align=left|{{sort|2|}} Partially resolved. A significant topic of research throughout the 20th century, resulting in solutions for some cases.<ref name="Gilbarg Trudinger 2001 Elliptic PDEs ">{{cite book | last1=Gilbarg | first1=David | last2=Trudinger | first2=Neil S. | title=Elliptic Partial Differential Equations of Second Order | publisher=Springer Science & Business Media | publication-place=Berlin New York | date=2001-01-12 | isbn=978-3-540-41160-4 | page=}}</ref><ref name="Serrin 1969 Dirichlet">{{cite journal | last= Serrin | first=James | title=The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables | journal=Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences | volume=264 | issue=1153 | date=1969-05-08 | issn=0080-4614 | doi=10.1098/rsta.1969.0033 | pages=413–496| bibcode=1969RSPTA.264..413S }}</ref><ref name="Mawhin 1999 pp. 195–228">{{cite journal | last=Mawhin | first=Jean | title=Leray-Schauder degree: a half century of extensions and applications | journal=Topological Methods in Nonlinear Analysis | publisher=Nicolaus Copernicus University in Toruń, Juliusz Schauder Center for Nonlinear Studies | volume=14 | issue=2 | date=1 January 1999 | issn=1230-3429 | doi=10.12775/TMNA.1999.029 | pages=195–228 | url=https://projecteuclid.org/journals/topological-methods-in-nonlinear-analysis/volume-14/issue-2/Leray-Schauder-degree--a-half-century-of-extensions-and/tmna/1475179840.full | access-date=8 April 2024}}</ref>}}
|style="text-align:center"| ?
|-
|style="text-align:center"| [[Hilbert's twenty-first problem|21st]]
| Proof of the existence of [[Fuchsian theory|Fuchsian]] [[linear differential equation]]s having a prescribed [[monodromy group]]
|{{Yes|{{sort|2|}} Resolved. Result: No, shown by A. Bolibrukh. <ref name="Plemelj">{{Citation | last1=Plemelj | first1=Josip | editor1-last=Radok. | editor1-first=J. R. M. | title=Problems in the sense of Riemann and Klein | url=https://books.google.com/books?id=f0urAAAAIAAJ | publisher=Interscience Publishers John Wiley & Sons Inc.|location= New York-London-Sydney | series= Interscience Tracts in Pure and Applied Mathematics | mr=0174815 | year=1964 | volume=16| isbn=9780470691250 }}</ref><ref name ="Anasov_Bolibruch">{{Citation | last1=Anosov | first1=D. V. | last2=Bolibruch | first2=A. A. | authorlink2=Andrei Bolibrukh | title=The Riemann-Hilbert problem | publisher=Friedr. Vieweg & Sohn | location=Braunschweig | series=Aspects of Mathematics, E22 | isbn=978-3-528-06496-9 | mr=1276272 | year=1994 | doi=10.1007/978-3-322-92909-9}}</ref><ref name ="Bolibruch1">{{Citation | last1=Bolibrukh | first1=A. A. | title=The Riemann-Hilbert problem | doi=10.1070/RM1990v045n02ABEH002350 | mr=1069347 | year=1990 | journal=Akademiya Nauk SSSR I Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk | issn=0042-1316 | volume=45 | issue=2 | pages=3–47| bibcode=1990RuMaS..45Q...1B | s2cid=250853546 |language=Russian}}</ref><ref name ="Bolibruch2">{{Citation | last1=Bolibrukh | first1=A.A. | title=Sufficient conditions for the positive solvability of the Riemann-Hilbert problem | journal=Matematicheskie Zametki |pages=110–117 | language=Russian | year=1992| volume=51 | issue=2 |mr=1165460 | doi=10.1007/BF02102113| s2cid = 121743184}}</ref>|align=left}}
|style="text-align:center"| 1989
|-
|style="text-align:center"| [[Hilbert's twenty-second problem|22nd]]
| Uniformization of analytic relations by means of [[automorphic function]]s
|{{partial|align=left|{{sort|2|}} Partially resolved. [[Uniformization theorem]]}}
|style="text-align:center"| ?
|-
|style="text-align:center"| [[Hilbert's twenty-third problem|23rd]]
| Further development of the [[calculus of variations]]
|{{unknown|align=left|style=font-size:inherit|{{sort|4|}} Too vague to be stated resolved or not.}}
|style="text-align:center"| —
|}