Hilbert's problems are a list of twenty-three problems in mathematics put forth by German mathematician David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900. The problems were all unsolved at the time, and several of them turned out to be very influential for 20th century mathematics. Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21 and 22) at the conference, speaking on 8 August in the Sorbonne; the full list was published later. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... | name = David Hilbert | image = Hilbert1912. ... This article is about the capital of France. ... The International Congress of Mathematicians (ICM) is the biggest congress in mathematics. ... Äž: For the film, see: 1900 (film). ... (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s As a means of recording the passage of time, the 20th century was that century which lasted from 1901–2000 in the sense of the Gregorian calendar (1900–1999 in the... is the 220th day of the year (221st in leap years) in the Gregorian calendar. ... The Sorbonne, Paris, in a 17th century engraving The historic University of Paris (French: ) first appeared in the second half of the 12th century, but was in 1970 reorganised as 13 autonomous universities (University of Paris I–XIII). ...

Nature and influence of the problems

Hilbert's problems ranged widely, not only across many areas of pure and applied mathematics, but also in scope and precision. Some of them are propounded precisely enough to admit a clear affirmative/negative answer, like the 3rd problem (probably the easiest for a nonspecialist to understand and also the first to be solved) or the notorious 8th problem (the Riemann hypothesis). There are other problems (notably the 5th) for which experts have traditionally agreed on a single interpretation and a solution to the accepted interpretation has been given, but for which there remain open problems which are so closely related so as to be, perhaps, part of what Hilbert had in mind. Sometimes Hilbert's statements were not precise enough to specify a particular problem but were suggestive enough so that certain problems of more contemporary origin seem to fit in nicely, e.g. most modern number theorists would probably see the 9th problem as referring to the (conjectural) Langlands correspondence on representations of the absolute Galois group of a number field. Still other problems (e.g. the 11th and the 16th) call for work on what are now flourishing mathematical subdisciplines, like the theories of quadratic forms and real algebraic curves. There is also the Riemann hypothesis for curves over finite fields. ... In mathematics, a Galois group is a group associated with a certain type of field extension. ... In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days...

There are two problems which are not only unresolved but may in fact be unresolvable by modern standards. The 6th problem calls for the axiomatization of physics, a goal that twentieth century developments in physics (including its recognition as a discipline independent from mathematics) seem to render both more remote and less important than in Hilbert's day. Also the 4th problem concerns the foundations of geometry, in a manner which is now generally judged to be too vague to admit a definitive answer.

Remarkably, the other twenty-one problems have all received significant attention, and late into the twentieth century work on these problems was still considered to be of the highest importance. Notably, Paul Cohen received the Fields Medal in 1966 for his work on the first problem, and the negative solution of the tenth problem in 1970 by Matiyasevich (completing work of Davis, Putnam and Robinson) generated similar excitement and acclaim. Aspects of these problems are still of great interest today. Paul Joseph Cohen (April 2, 1934 – March 23, 2007[1]) was an American mathematician. ... The obverse of the Fields Medal The Fields Medal is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union, a meeting that takes place every four years. ... Martin Davis, (born 1926, New York City) is an American mathematician, known for his work on Hilberts tenth problem. ... Hilary Whitehall Putnam (born July 31, 1926) is an American philosopher who has been a central figure in Western philosophy since the 1960s, especially in philosophy of mind, philosophy of language, and philosophy of science. ... Julia Hall Bowman Robinson (December 8, 1919 - July 30, 1985) was an American mathematician, born in Saint Louis, Missouri. ...

Ignorabimus

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Several of the Hilbert problems have been resolved (or arguably resolved) in ways that would have been profoundly surprising, and even disturbing, to Hilbert himself. Following Frege and Russell, Hilbert sought to place mathematics on a sound logical foundation using the method of formal systems, i.e., finitistic proofs from an agreed upon set of axioms. One of the main goals of Hilbert's program was a finitistic proof of the consistency of the axioms of arithmetic: that is his second problem. Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, IPA: ) was a German mathematician who became a logician and philosopher. ... Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS, (18 May 1872 – 2 February 1970), was a British philosopher, logician, mathematician, advocate for social reform, and pacifist. ... In logic and mathematics, a formal system consists of two components, a formal language plus a set of inference rules or transformation rules. ... Hilberts program, formulated by German mathematician David Hilbert in the 1920s, was to formalize all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent. ...

However, Gödel's second incompleteness theorem gives a precise sense in which such a finitistic proof of the consistency of arithmetic is provably impossible. Hilbert lived for 12 years after Gödel's theorem, but he does not seem to have written any formal response to or acknowledgement of Gödel's work. But there is no doubt that the significance of Gödel's work to mathematics as a whole (and not just the field of formal logic) was amply and dramatically illustrated by its applicability to one of Hilbert's problems. Kurt Gödel (IPA: ) (April 28, 1906 Brünn, Austria-Hungary (now Brno, Czech Republic) – January 14, 1978 Princeton, New Jersey) was an Austrian American mathematician and philosopher. ... In mathematical logic, Gödels incompleteness theorems, proved by Kurt Gödel in 1931, are two theorems stating inherent limitations of all but the most trivial formal systems for arithmetic of mathematical interest. ...

Hilbert's tenth problem does not ask whether there exists an algorithm for deciding the solvability of Diophantine equations but rather asks for the construction of such an algorithm: "to devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers." That the solution to this problem comes by way of showing that there can be no such algorithm would presumably have been very surprising to him.

On the other hand, in discussing his opinions that every mathematical problem should have a solution, Hilbert allows for the possibility that the solution could be a proof that the original problem is impossible. Famously, he stated that the point is to know one way or the other what the solution is, and he believed that we always can know this, that in mathematics there is no ignorabimus. It seems unclear whether he would have regarded the solution of the tenth problem as an instance of ignorabimus: what we are proving not to exist is not the integer solution, but (in a certain sense) our own ability to discern whether a solution exists. The ignorabimus, short for the Latin tag ignoramus et ignorabimus meaning we do not know and will not know, stood for a pessimistic (in one sense) position on the limits on scientific knowledge, in the thought of the nineteenth century. ...

On the other hand, the status of the first and second problems is even more complicated than this: there is no clear mathematical consensus as to whether the results of Gödel (in the case of the second problem) or Gödel and Cohen (in the case of the first problem) give definitive negative solutions or not, since these solutions apply to a certain formalization of the problems, a formalization which is quite reasonable but is not necessarily the only possible one. So the true status of, say, Hilbert's first problem is that some experts think that we know that we will never know the answer, but others think otherwise!^[who?] Although certain leading contemporary mathematicians^[who?] are working on the continuum hypothesis, it is not clear that their work is of a nature that will ever be convincing to all experts^{[weasel words]}. So although the important work of Gödel and Cohen is not at all in dispute, we may never reach consensus on whether it gives a solution to Hilbert's first problem.^{[weasel words]} It is hard to see how to avoid calling this ignorabimus.^{[weasel words]}

A round two dozen

Hilbert originally included 24 problems on his list, but decided against including one of them in the published list. The "24th problem" (in proof theory, on a criterion for simplicity and general methods) was rediscovered in Hilbert's original manuscript notes by German historian Rüdiger Thiele in 2000. Wikiquote has a collection of quotations related to: Simplicity Simplicity is the property, condition, or quality of being simple or un-combined. ...

Sequels

Since 1900, other mathematicians and mathematical organizations have announced problem lists, but, with few exceptions, these later problem collections have not had nearly as much influence or generated as much work as Hilbert's problems.

One of the exceptions is furnished by three conjectures made by André Weil in the late 1940s (the Weil conjectures). In the fields of algebraic geometry, number theory and the links between the two, the importance of the Weil conjectures was (and is) colossal. The first of the Weil conjectures was proved by Bernard Dwork, and a completely different proof of the first two conjectures via l-adic cohomology was given by Alexander Grothendieck. The last and deepest of the Weil conjectures (an analogue of the Riemann hypothesis) was proven by Pierre Deligne in what is widely regarded as one of the greatest mathematical achievements of all time. Both Grothendieck and Deligne were awarded the Fields medal. However, the Weil conjectures in their scope are more like a single Hilbert problem, and Weil never intended them as a programme for all mathematics. This is somewhat ironic, since arguably Weil was the mathematician of the 1940s and 1950s who best played the Hilbert role, being conversant with nearly all areas of (theoretical) mathematics and having played key roles in the development of many of them. André Weil (May 6, 1906 - August 6, 1998) (pronounced [1]) was one of the greatest mathematicians of the 20th century, whether measured by his research work, its influence on future work, exposition or breadth. ... In mathematics, the Weil conjectures, which had become theorems by 1974, were some highly-influential proposals from the late 1940s by André Weil on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields. ... Bernard Morris Dwork (May 27, 1923 – May 9, 1998) was an American mathematician, known for his application of p-adic analysis to local zeta functions, and in particular for the first general results on the Weil conjectures. ... Alexander Grothendieck (born March 28, 1928 in Berlin, Germany) is one of the most important mathematicians of the 20th century. ... Pierre Deligne, March 2005 Pierre Deligne (born 3 October 1944) is a Belgian mathematician. ... The obverse of the Fields Medal The Fields Medal is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union, a meeting that takes place every four years. ...

The turn of the millennium, being also the centennial of Hilbert's announcement of his problems, was a natural occasion to propose "a new set of Hilbert problems." Several mathematicians took up the challenge, notably Fields Medalist Steve Smale, who responded to a request of Vladimir Arnold by proposing a list of 18 problems. Smale's problems have thus far not received much attention from the media, and it seems unclear how much attention they are getting from the mathematical community. Stephen Smale (born July 15, 1930) is an American mathematician from Flint, Michigan, and winner of the Fields Medal in 1966. ... Vladimir Igorevich Arnold (Russian: Влади́мир И́горевич Арно́льд, born June 12, 1937 in Odessa, USSR) is one of the worlds most prolific mathematicians. ... Smales problems refers to a list of eighteen unsolved problems in mathematics, proposed by Steve Smale in 2000. ...

At least in the mainstream media, the de facto 21st century analogue of Hilbert's problems is the list of seven Millennium Prize Problems chosen in 2000 by the Clay Mathematics Institute. Unlike the Hilbert problems, where the primary award was the admiration of Hilbert in particular and the worldwide mathematical community in general, each prize problem carries a million dollar bounty. As for the Hilbert problems, one of the prize problems (the Poincare conjecture) was solved relatively soon after the problems were announced. The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Cambridge, Massachusetts, and dedicated to increasing and disseminating mathematical knowledge. ... The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Cambridge, Massachusetts. ... In mathematics, the Poincaré conjecture is a conjecture about the characterisation of the three-dimensional sphere amongst 3-manifolds. ...

Noteworthy for its appearance on the list of Hilbert problems, Smale's list and the list of Millennium Prize Problems -- and even, in its geometric guise, in the Weil Conjectures -- is the Riemann hypothesis. Notwithstanding some famous (and occasionally loud) recent assaults from leading mathematicians of our day, many experts believe that the Riemann hypothesis will lurk at the top of problem lists for some centuries yet to come. The maestro himself had this to say: "If I were to awaken after having slept for a thousand years, my first question would be: has the Riemann hypothesis been proven?" — David Hilbert

Summary

Of the cleanly-formulated Hilbert problems, problems 3, 7, 10, 11, 13, 14, 17, 19, 20, and 21 have a resolution that is accepted by consensus. On the other hand, problems 1, 2, 5, 9, 12, 15, 18⁺, and 22 have solutions that have partial acceptance, but where there exists some controversy as to whether it resolves the problem.

The + on 18 denotes that the Kepler conjecture solution is a computer-assisted proof, a notion anachronistic for a Hilbert problem and also to some extent controversial because of its lack of verifiability by a human reader in a reasonable time. In mathematics, the Kepler conjecture is a conjecture about sphere packing in three dimensional Euclidean space. ... A computer-assisted proof is a mathematical proof that has been generated by computer. ...

That leaves 8 (the Riemann hypothesis) and 12 unresolved, both being in number theory. On this classification 4, 6, 16, and 23 are too loose to be ever described as solved. The withdrawn 24 would also fall in this class. There is also the Riemann hypothesis for curves over finite fields. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ...

Table of problems

Hilbert's twenty-three problems are:

In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... Not to be confused with Natural number. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ... In mathematical logic, a formal system is consistent if it does not contain a contradiction, or, more precisely, for no proposition are both and provable. ... This article is about a logical statement. ... Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. ... In mathematical logic, a formal system is consistent if it does not contain a contradiction, or, more precisely, for no proposition φ are both φ and ¬φ provable. ... In mathematical logic, Gödels incompleteness theorems, proved by Kurt Gödel in 1931, are two theorems stating inherent limitations of all but the most trivial formal systems for arithmetic of mathematical interest. ... In mathematics, ε0 is the smallest transfinite ordinal number which cannot be reached from ω (the smallest transfinite ordinal) with a finite number of the ordinal operations of addition, multiplication and exponentiation. ... The third on Hilberts list of mathematical problems, presented in 1900, is the easiest one. ... In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ... The third on Hilberts list of mathematical problems, presented in 1900, is the easiest one. ... In mathematics, Hilberts fourth problem in the 1900 Hilbert problems was a foundational question in geometry. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the space. ... In mathematics, a Lie group (IPA ) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... This picture illustrates how the hours on a clock form a group under modular addition. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... Andrew Mattei Gleason is an American mathematician. ... In mathematics, the Hilbert-Smith conjecture is concerned with the transformation groups of manifolds; and in particular with the limitations on topological groups G that can act effectively (faithfully) on a (topological) manifold M. Restricting to G which are locally compact and have a continuous, faithful group action on M... Hilberts sixth problem is to axiomatize those branches of science in which mathematics is prevalent. ... This article is about a logical statement. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Hilberts seventh problem concerns the irrationality and transcendence of certain numbers (Irrationalität und Transzendenz bestimmter Zahlen). ... In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ... In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ... In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. ... In mathematics, the Gelfond-Schneider theorem is the following statement, originally proved by Aleksandr Gelfond: If is an algebraic number (with and ), and is an irrational algebraic number, then is a transcendental number. ... In mathematics, the Gelfond-Schneider theorem is the following statement, originally proved by Aleksandr Gelfond: If is an algebraic number (with and ), and is an irrational algebraic number, then is a transcendental number. ... In mathematics, the Riemann hypothesis (aka Riemann zeta hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous of all unsolved problems. ... There is also the Riemann hypothesis for curves over finite fields. ... In mathematics, the term trivial is frequently used for objects (for examples, groups or topological spaces) that have a very simple structure. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... Goldbachs conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. ... Look up een, even in Wiktionary, the free dictionary. ... In mathematics, a prime number (or a prime) is a natural number greater than 1 which has exactly two distinct natural number divisors: 1 and itself. ... In mathematics, Hilberts ninth problem was to find the most general law of reciprocity in an algebraic number field. ... In number theory, the law of quadratic reciprocity connects the solvability of two related quadratic equations in modular arithmetic. ... This article is about the branch of mathematics. ... In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days... Hilberts tenth problem is the tenth on the list of Hilberts problems of 1900. ... In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. ... Matiyasevichs theorem, proven in 1970 by Yuri Matiyasevich, implies that Hilberts tenth problem is unsolvable. ... Hilberts eleventh problem, a furthering of the theory of quadratic forms, was stated thus in his landmark speech: It is considered to have been addressed by Helmut Hasses principles in 1923 and 1924. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... For other senses of this word, see coefficient (disambiguation). ... Hilberts twelfth problem, of the 23 Hilberts problems, is the extension of Kroneckers Theorem on abelian fields to any algebraic realm of rationality. ... In abstract algebra, an abelian extension is a field extension for which the associated Galois group is abelian. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days... 21. ... The degree of a polynomial is the maximum of the degrees of all terms in the polynomial. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... The factual accuracy of this article is disputed. ... Andrey Nikolaevich Kolmogorov (Андре́й Никола́евич Колмого́ров) (kahl-mah-GAW-raff) (April 25, 1903 in Tambov - October 20, 1987 in Moscow) was a Russian mathematician who made major advances in the fields of probability theory and topology. ... Vladimir Igorevich Arnold (Russian: Влади́мир И́горевич Арно́льд, born June 12, 1937 in Odessa, USSR) is one of the worlds most prolific mathematicians. ... Hilberts fourteenth problem asks whether certain subrings are finitely generated. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... Hilberts fifteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. ... In mathematics, Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry (part of enumerative geometry). ... Hilberts sixteenth problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900, together with the other 22 problems. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... Hilberts seventeenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. ... In grammatical theory, definiteness is a feature of noun phrases, distinguishing between entities which are specific and identifiable in a given context (definite noun phrases) and entities which are not (indefinite noun phrases). ... In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ... In mathematics, a quotient is the end result of a division problem. ... Addition is one of the basic operations of arithmetic. ... In algebra, the square of a number is that number multiplied by itself. ... Hilberts eighteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. ... A tessellated plane seen in street pavement. ... For the game magazine, see Polyhedron (magazine). ... Sphere packing finds practical application in the stacking of oranges. ... Hilberts nineteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... Analytic may refer to Analytic proposition or analytic philosophy, in philosophy Analytic geometry, analytic function, analytic continuation, analytic set in mathematics. ... Ennio de Giorgi (1928 - 1996) was an Italian mathematician. ... John Forbes Nash, Jr. ... Hilberts twentieth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. ... Calculus of variations is a field of mathematics that deals with functionals, as opposed to ordinary calculus which deals with functions. ... In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ... The twenty-first problem of the 23 Hilbert problems, from the celebrated list put forth in 1900 by David Hilbert, was phrased like this (English translation from 1902). ... In mathematics, a linear differential equation is a differential equation of the form Ly = f, where the differential operator L is a linear operator, y is the unknown function, and the right hand side f is a given function. ... In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and differential geometry behave as they run round a singularity. ... Hilberts twenty-second problem is the penultimate entry in the celebrated list of 23 Hilbert problems compiled in 1900 by David Hilbert. ... In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms. ... Hilberts twenty-third problem is the last of Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. ... Calculus of variations is a field of mathematics that deals with functionals, as opposed to ordinary calculus which deals with functions. ...

Notes

1. ^ According to Rowe & Gray (see reference below), most of the problems have been solved. Some were not completely defined, but enough progress has been made to consider them "solved"; Rowe & Gray lists the fourth problem as too vague to say whether it has been solved.
2. ^ Problem 9 has been solved in the abelian case, by the development of class field theory; the non-abelian case remains unsolved, if one interprets that as meaning non-abelian class field theory.
3. ^ Shreeram S. Abhyankar: Hilbert’s Thirteenth Problem (last retrieved on March 20, 2008)
4. ^ A. G. Vitushkin: On Hilbert’s thirteenth problem and related questions, (last retrieved on March 20, 2008)
5. ^ N. G. Chebotarev, “On certain questions of the problem of resolvents”
6. ^ D. Hilbert, “¨Uber die Gleichung neunten Grades”, Math. Ann. 97 (1927), 243–250
7. ^ Rowe & Gray also list the 18th problem as "open" in their 2000 book, because the sphere-packing problem (also known as the Kepler conjecture) was unsolved, but a solution to it has now been claimed (see reference below).

Abelian, in mathematics, is used in many different definitions: In group theory: Abelian group, a group in which the binary operation is commutative Category of abelian groups Ab has abelian groups as objects and group homomorphisms as morphisms Metabelian group is a group where the commutator subgroup is contained in... In mathematics, class field theory is a major branch of algebraic number 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. While class field theory was essentially known... In mathematics, the Kepler conjecture is a conjecture about sphere packing in three dimensional Euclidean space. ...

External links

• Listing of the 23 problems, with descriptions of which have been solved
• Original text of Hilbert's talk, in German
• English translation of Hilbert's 1900 address
• Details on the solution of the 18th problem
• "On Hilbert's 24th Problem: Report on a New Source and Some Remarks."
• The Paris Problems
• Hilbert's Tenth Problem page!

References

• Rowe, David; Gray, Jeremy J. (2000). The Hilbert Challenge. Oxford University Press. ISBN 0-19-850651-1
• Yandell, Benjamin H. (2002). The Honors Class. Hilbert's Problems and Their Solvers. A K Peters. ISBN 1-56881-141-1
• On Hilbert and his 24 Problems. In: Proceedings of the Joint Meeting of the CSHPM 13(2002)1-22 (26th Meeting; ed. M. Kinyon)
• John Dawson, Jr Logical Dilemmas, The Life and Work of Kurt Gödel, AK Peters, Wellesley, Mass., 1997. A wealth of information relevant to Hilbert's "program" and Gödel's impact on the Second Question, the impact of Arend Heyting's and Brouwer's Intuitionism on Hilbert's philosophy. Dawson is Professor of Mathematics at Penn State U, cataloguer of Gödel's papers for the Institute for Advanced Study in Princeton, and a co-editor of Gödel's Collected Works.
• Felix E. Browder (editor), Mathematical Developments Arising from Hilbert Problems, Proceedings of Symposia in Pure Mathematics XXVIII (1976), American Mathematical Society. A collection of survey essays by experts devoted to each of the 23 problems emphasizing current developments.
• Yuri Matiyasevich, Hilbert's Tenth Problem, MIT Press, Cambridge, Massachusetts, 1993. An account at the undergraduate level by the mathematican who completed the solution of the problem.