Hilbert's problems
Hilbert's problems form a list of twenty-three problems in mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 published by German
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

 David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

 in 1900. The problems were all unsolved
Unsolved problems in mathematics
This article lists some unsolved problems in mathematics. See individual articles for details and sources.- Millennium Prize Problems :Of the seven Millennium Prize Problems set by the Clay Mathematics Institute, six have yet to be solved:* P versus NP...

 at the time, and several of them were 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 Paris
Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...

 conference of the International Congress of Mathematicians
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....

, speaking on 8 August in the Sorbonne
University of Paris
The University of Paris was a university located in Paris, France and one of the earliest to be established in Europe. It was founded in the mid 12th century, and officially recognized as a university probably between 1160 and 1250...

. The complete list of 23 problems was published later, most notably in English translation in 1902 by Mary Frances Winston Newson
Mary Frances Winston Newson
Mary Frances Winston Newson was an American mathematician. She became the first female American to receive a PhD in mathematics from a European university, namely the University of Göttingen in Germany....

 in the Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
The Bulletin of the American Mathematical Society is a quarterly mathematical journal published by the American Mathematical Society...


Nature and influence of the problems

Hilbert's problems ranged greatly in topic and precision. Some of them are propounded precisely enough to enable 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
Riemann hypothesis
In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...

). 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 unsolved problems which are so closely related as to be, perhaps, part of what Hilbert intended. 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 apply, 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
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...

 of a number field. Still other problems (e.g. the 11th and the 16th) concern what are now flourishing mathematical subdisciplines, like the theories of quadratic forms and real algebraic curves.

There are two problems which are not only unresolved but may in fact be unresolvable by modern standards. The 6th problem concerns the axiomatization of physics, a goal that twentieth century developments of physics (including its recognition as a discipline independent from mathematics) seem to render both more remote and less important than in Hilbert's time. Also, the 4th problem concerns the foundations of geometry, in a manner which is now generally judged to be too vague to enable 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 greatest importance. Notably, Paul Cohen
Paul Cohen (mathematician)
Paul Joseph Cohen was an American mathematician best known for his proof of the independence of the continuum hypothesis and the axiom of choice from Zermelo–Fraenkel set theory, the most widely accepted axiomatization of set theory.-Early years:Cohen was born in Long Branch, New Jersey, into a...

 received the Fields Medal
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, 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...

 during 1966 for his work on the first problem, and the negative solution of the tenth problem during 1970 by Yuri Matiyasevich
Yuri Matiyasevich
Yuri Vladimirovich Matiyasevich, is a Russian mathematician and computer scientist. He is best known for his negative solution of Hilbert's tenth problem, presented in his doctoral thesis, at LOMI .- Biography :* In 1962-1963 studied at Saint Petersburg Lyceum 239...

 (completing work of Martin Davis
Martin Davis
Martin David Davis, is an American mathematician, known for his work on Hilbert's tenth problem . He received his Ph.D. from Princeton University in 1950, where his adviser was Alonzo Church . He is Professor Emeritus at New York University. He is the co-inventor of the Davis-Putnam and the DPLL...

, Hilary Putnam
Hilary Putnam
Hilary Whitehall Putnam is an American philosopher, mathematician and computer scientist, who has been a central figure in analytic philosophy since the 1960s, especially in philosophy of mind, philosophy of language, philosophy of mathematics, and philosophy of science...

 and Julia Robinson
Julia Robinson
Julia Hall Bowman Robinson was an American mathematician best known for her work on decision problems and Hilbert's Tenth Problem.-Background and education:...

) generated similar acclaim. Aspects of these problems are still of great interest today.


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
Gottlob Frege
Friedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...

 and Russell
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...

, Hilbert sought to define mathematics logically using the method of formal system
Formal system
In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...

s, i.e., finitistic
In the philosophy of mathematics, one of the varieties of finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps...

 proofs from an agreed-upon set of axioms. One of the main goals of Hilbert's program
Hilbert's program
In mathematics, Hilbert's program, formulated by German mathematician David Hilbert, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies...

 was a finitistic proof of the consistency of the axioms of arithmetic: that is his second problem.

However, Gödel's
Kurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...

 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 Gödel's work. But doubtless the significance of Gödel's work to mathematics as a whole (and not just to formal logic) was amply and dramatically illustrated by its applicability to one of Hilbert's problems.

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 this problem was solved by showing that there cannot be any such algorithm would presumably have been very surprising to him.

In discussing his opinion 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 not any "ignorabimus
The Latin maxim ignoramus et ignorabimus, meaning "we do not know and will not know", stood for a position on the limits of scientific knowledge, in the thought of the nineteenth century...

" (statement that the truth can never be known). 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.

On the other hand, the status of the first and second problems is even more complicated: there is not any 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.

The 24th Problem

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
Simplicity is the state or quality of being simple. It usually relates to the burden which a thing puts on someone trying to explain or understand it. Something which is easy to understand or explain is simple, in contrast to something complicated...

 and general methods) was rediscovered in Hilbert's original manuscript notes by German historian Rüdiger Thiele in 2000.


Since 1900, other mathematicians and mathematical organizations have announced problem lists, but, with few exceptions, these collections have not had nearly as much influence nor generated as much work as Hilbert's problems.

One of the exceptions is furnished by three conjectures made by André Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...

 during the late 1940s (the Weil conjectures
Weil conjectures
In mathematics, the Weil conjectures were some highly-influential proposals by on the generating functions derived from counting the number of points on algebraic varieties over finite fields....

). In the fields of algebraic geometry, number theory and the links between the two, the Weil conjectures were very important . The first of the Weil conjectures was proved by Bernard Dwork
Bernard Dwork
Bernard Morris Dwork 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. Together with Kenkichi Iwasawa he received the Cole Prize in 1962.Dwork received his Ph.D. at Columbia...

, and a completely different proof of the first two conjectures via l-adic cohomology
Étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...

 was given by Alexander Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...

. The last and deepest of the Weil conjectures (an analogue of the Riemann hypothesis) was proven by Pierre Deligne
Pierre Deligne
- See also :* Deligne conjecture* Deligne–Mumford moduli space of curves* Deligne–Mumford stacks* Deligne cohomology* Fourier–Deligne transform* Langlands–Deligne local constant- External links :...

. Both Grothendieck and Deligne were awarded the Fields medal
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, 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...

. 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 been important in the development of many of them.

Paul Erdős
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...

 is legendary for having posed hundreds, if not thousands, of mathematical problems, many of them profound. Erdős often offered monetary rewards; the size of the reward depended on the perceived difficulty of the problem.

The end 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 accepted the challenge, notably Fields Medalist Steve Smale, who responded to a request of Vladimir Arnold
Vladimir Arnold
Vladimir Igorevich Arnold was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable Hamiltonian systems, he made important contributions in several areas including dynamical systems theory, catastrophe theory,...

 by proposing a list of 18 problems.
Smale's problems
Smale's problems
Smale's problems refers to a list of eighteen unsolved problems in mathematics that was proposed by Steve Smale in 2000. Smale composed this list in reply to a request from Vladimir Arnold, then president of the International Mathematical Union, who asked several mathematicians to propose a list of...

 have thus far not received much attention from the media, and it is unclear how much serious attention they are getting from the mathematical community.

At least in the mainstream media, the de facto 21st century analogue of Hilbert's problems is the list of seven Millennium Prize Problems
Millennium Prize Problems
The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. As of September 2011, six of the problems remain unsolved. A correct solution to any of the problems results in a US$1,000,000 prize being awarded by the institute...

 chosen during 2000 by the Clay Mathematics Institute
Clay Mathematics Institute
The Clay Mathematics Institute is a private, non-profit foundation, based in Cambridge, Massachusetts. The Institute is dedicated to increasing and disseminating mathematical knowledge. It gives out various awards and sponsorships to promising mathematicians. The institute was founded in 1998...

. Unlike the Hilbert problems, where the primary award was the admiration of Hilbert in particular and mathematicians in general, each prize problem includes a million dollar bounty. As with the Hilbert problems, one of the prize problems (the Poincaré conjecture
Poincaré conjecture
In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...

) was solved relatively soon after the problems were announced.

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 recent assaults from major mathematicians of our day, many experts believe that the Riemann hypothesis will be included in problem lists for centuries yet. Hilbert himself declared: "If I were to awaken after having slept for a thousand years, my first question would be: has the Riemann hypothesis been proven?"

During 2008, DARPA announced its own list of 23 problems which it hoped could cause major mathematical breakthroughs, "thereby strengthening the scientific and technological capabilities of DoD".


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, 15, 18+, and 22 have solutions that have partial acceptance, but there exists some controversy as to whether it resolves the problem.

The + on 18 denotes that the Kepler conjecture
Kepler conjecture
The Kepler conjecture, named after the 17th-century German astronomer Johannes Kepler, is a mathematical conjecture about sphere packing in three-dimensional Euclidean space. It says that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic...

 solution is a computer-assisted proof
Computer-assisted proof
A computer-assisted proof is a mathematical proof that has been at least partially generated by computer.Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of a mathematical theorem. The idea is to use a computer program to perform lengthy computations, and...

, a notion anachronistic for a Hilbert problem and to some extent controversial because of its lack of verifiability by a human reader in a reasonable time.

That leaves 16, 8 (the Riemann hypothesis
Riemann hypothesis
In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...

) and 12 unresolved. On this classification 4, 16, and 23 are too vague to ever be described as solved. The withdrawn 24 would also be in this class. 6 is considered as a problem in physics rather than in mathematics.

Table of problems

Hilbert's twenty-three problems are:
Problem Brief explanation Status Year Solved
1st The continuum hypothesis
Continuum hypothesis
In mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...

 (that is, there is no set whose cardinality is strictly between that of the integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s and that of the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

Hilbert's second problem
In mathematics, Hilbert's second problem was posed by David Hilbert in 1900 as one of his 23 problems. It asks for a proof that arithmetic is consistent – free of any internal contradictions....

Prove that the axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

s of arithmetic
Arithmetic or arithmetics 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. It involves the study of quantity, especially as the result of combining numbers...

 are consistent
Consistency proof
In logic, a consistent theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if and only if it has a model, i.e. there exists an interpretation under which all...

Hilbert's third problem
The third on Hilbert's list of mathematical problems, presented in 1900, is the easiest one. The problem is related to the following question: given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the...

Given any two polyhedra
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

 of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second?
Hilbert's fourth problem
In mathematics, Hilbert's fourth problem in the 1900 Hilbert problems was a foundational question in geometry. In one statement derived from the original, it was to find geometries whose axioms are closest to those of Euclidean geometry if the ordering and incidence axioms are retained, the...

Construct all metric
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

s where lines are geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

Hilbert's fifth problem
Hilbert's fifth problem, is the fifth mathematical problem from the problem-list publicized in 1900 by mathematician David Hilbert, and concerns the characterization of Lie groups. The theory of Lie groups describes continuous symmetry in mathematics; its importance there and in theoretical physics...

Are continuous groups
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 automatically differential groups
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

Hilbert's sixth problem
Hilbert's sixth problem is to axiomatize those branches of physics in which mathematics is prevalent. It occurs on the widely cited list of Hilbert's problems in mathematics that he presented in the year 1900...

In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

atize all of physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

Hilbert's seventh problem
Hilbert's seventh problem is one of David Hilbert's list of open mathematical problems posed in 1900. It concerns the irrationality and transcendence of certain numbers...

Is a b transcendental
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

, for algebraic
Algebraic number
In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...

 a ≠ 0,1 and irrational
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

 algebraic b ?
Hilbert's eighth problem
Hilbert's eighth problem is one of David Hilbert's list of open mathematical problems posed in 1900. It concerns number theory, and in particular the Riemann hypothesis, although it is also concerned with the Goldbach Conjecture...

The Riemann hypothesis
Riemann hypothesis
In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...

 ("the real part of any non-trivial
Trivial (mathematics)
In mathematics, the adjective trivial is frequently used for objects that have a very simple structure...

 zero of the Riemann zeta function is ½") and other prime number problems, among them Goldbach's conjecture
Goldbach's conjecture
Goldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. It states:A Goldbach number is a number that can be expressed as the sum of two odd primes...

 and the twin prime conjecture
Hilbert's ninth problem
Hilbert's ninth problem, from the list of 23 Hilbert's problems , 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.- Progress made :...

Find most general law of the reciprocity theorem
Quadratic reciprocity
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic which gives conditions for the solvability of quadratic equations modulo prime numbers...

 in any algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

ic number field
Hilbert's tenth problem
Hilbert's tenth problem is the tenth on the list of Hilbert's problems of 1900. Its statement is as follows:Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite...

Find an algorithm to determine whether a given polynomial Diophantine equation
Diophantine equation
In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...

 with integer coefficients has an integer solution.
Hilbert's eleventh problem
Hilbert's eleventh problem is one of David Hilbert's list of open mathematical problems posed in 1900. A furthering of the theory of quadratic forms, he stated the problem as follows:...

Solving quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....

s with algebraic numerical coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...

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...

Extend the Kronecker–Weber theorem
Kronecker–Weber theorem
In algebraic number theory, the Kronecker–Weber theorem states that every finite abelian extension of the field of rational numbers Q, or in other words, every algebraic number field whose Galois group over Q is abelian, is a subfield of a cyclotomic field, i.e. a field obtained by adjoining a root...

 on abelian extensions of the rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

s to any base number field.
Hilbert's thirteenth problem
Hilbert's thirteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It entails proving whether or not a solution exists for all 7th-degree equations using functions of two arguments...

Partially solved 7-th degree equations
Degree of a polynomial
The degree of a polynomial represents the highest degree of a polynominal's terms , should the polynomial be expressed in canonical form . The degree of an individual term is the sum of the exponents acting on the term's variables...

 using continuous functions of two parameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....

Hilbert's fourteenth problem
In mathematics, Hilbert's fourteenth problem, that is, number 14 of Hilbert's problems proposed in 1900, asks whether certain rings are finitely generated....

Is the ring of invariants
Invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...

 of an algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...

 acting on a polynomial ring
Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...

 always finitely generated
Finitely generated algebra
In mathematics, a finitely generated algebra is an associative algebra A over a field K where there exists a finite set of elements a1,…,an of A such that every element of A can be expressed as a polynomial in a1,…,an, with coefficients in K...

Hilbert's fifteenth problem
Hilbert's fifteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It entails a rigorous foundation of Schubert's enumerative calculus....

Rigorous foundation of Schubert's enumerative calculus.
Hilbert's sixteenth problem
Hilbert's 16th problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900, together with the other 22 problems....

Describe relative positions of ovals originating from a real algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...

 and as limit cycles of a polynomial vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

 on the plane.
Hilbert's seventeenth problem
Hilbert's seventeenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It entails expression of definite rational functions as quotients of sums of squares...

Expression of definite rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...

 as quotient
In mathematics, a quotient is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient further is expressed as the number of times the divisor divides into the dividend e.g. The quotient of 6 and 2 is also 3.A...

 of sums of squares
Hilbert's eighteenth problem
Hilbert's eighteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by mathematician David Hilbert. It asks three separate questions about lattices and sphere packing in Euclidean space....

(a) Is there a polyhedron which admits only an anisohedral tiling
Anisohedral tiling
In geometry, a shape is said to be anisohedral if it admits a tiling, but no such tiling is isohedral ; that is, in any tiling by that shape there are two tiles that are not equivalent under any symmetry of the tiling...

 in three dimensions?
(b) What is the densest sphere packing
Sphere packing
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space...

(a) 1928
(b) 1998
Hilbert's nineteenth problem
Hilbert's nineteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It asks whether the solutions of regular problems in the calculus of variations are always analytic.-History:...

Are the solutions of Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...

s always analytic
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...

Hilbert's twentieth problem
Hilbert's twentieth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It asks whether all boundary value problems can be solved...

Do all variational problems
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...

 with certain boundary conditions have solutions?
Hilbert's twenty-first 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 ....

Proof of the existence of linear differential equation
Linear differential equation
Linear differential equations are of the formwhere the differential operator L is a linear operator, y is the unknown function , and the right hand side ƒ is a given function of the same nature as y...

s having a prescribed monodromic group
Hilbert's twenty-second problem
Hilbert's twenty-second problem is the penultimate entry in the celebrated list of 23 Hilbert problems compiled in 1900 by David Hilbert. It entails the uniformization of analytic relations by means of automorphic functions....

Uniformization of analytic relations by means of automorphic function
Automorphic function
In mathematics, an automorphic function is a function on a space that is invariant under the action of some group, in other words a function on the quotient space. Often the space is a complex manifold and the group is a discrete group....

Hilbert's twenty-third problem
Hilbert's twenty-third problem is the last of Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. In contrast with Hilbert's other 22 problems, his 23rd is not so much a specific "problem" as an encouragement towards further development of the calculus of variations...

Further development of the calculus of variations

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