At the time of the second international Congress of mathematics held with Paris in 1900, David Hilbert presented a list of problems which held the mathematicians in failure hitherto. These problems had, according to Hilbert, to mark the course of mathematics of the 20th century, and one can say today that was largely the case. The final list was published after the behavior of the congress and is today familiarly called problems of Hilbert .

The following sections briefly present each problem.

The equivalent for the 21e century was defined under the program of the Problèmes of the price of the millenium.

First problem

See also: Assumption of continuous the

Prouver the Assumption of continuous the of Cantor.

Paul Cohen, while being based on work of Gödel, showed in 1963 that this conjecture was Indécidable .

Hilbert attaches this problem to the following question: to prove that the whole of the real numbers can be well ordered. Ernst Zermelo proved that the existence of this Bon order is equivalent to the Axiome of the choice of Zermelo. Thus, to prove it come down to accept this axiom, which many mathematicians refused. Whereas Hilbert thought that these two problems were dependant, Cohen proved that they were independent by showing that the assumption of continuous of Cantor was indécidable.

Second problem

See also: Second problem of Hilbert

To show the consistency of the Axiom S of the Arithmetic .

Gödel showed in 1931, via its theorem of incomplétude, that could not be shown without leaving the arithmetic one. Gerhard Gentzen, however, gave, in 1936, an affirmative response by means of a transfinite recurrence.

Third problem

See also: Third problem of Hilbert

Étant given two Polyhedral S of equal Volume, can one cut out the first polyhedron in polyhedrons and gather them to form the second polyhedron?

max Dehn, raises of Hilbert, showed that not, in 1902, by showing that it was impossible to divide a cube and a regular tetrahedron of the same volume into a finished number of polyhedrons two to two identical. Despite everything, the Paradoxe of Banach-Tarski constitutes a positive test for this question if it is not required that the intermediate pieces be polyhedrons and especially if one supposes the Axiome of the choice.

Fourth problem

See also: Fourth problem of Hilbert

to Définir all the Geometry S whose short distance between two points is a segment of droite.

The differential Géométrie made it possible to answer this problem partly, although one cannot strictly speaking of firm response.

Fifth problem

See also: Fifth problem of Hilbert

Démontrer that the groups of Dregs are necessarily différentiables.

The Théorème of Gleason-Montgomery-Zippin in 1953 answered it by the affirmative.

Sixth problem

See also: Sixth problem of Hilbert

L' axiomatization, founded on the mathematical model, of the Physical .

Because of appearance of the Theory of relativity and quantum Mechanical , the problem was quickly obsolete. Despite everything, one can note that the theoretical physics and mathematics do not cease approaching. By axiomatizing the Theory of probability, Kolmogorov solved this problem partly.

Seventh problem

See also: Seventh problem of Hilbert

Démontrer the transcendence of the $a^b$ numbers, with $a$ algebraic and $b$ irrational (for example $2^\; \{\backslash \; sqrt\; \{2\}\})$.

Work of Gelfond, supplemented by Schneider and Baker, made it possible to solve this problem partly (see Théorème of Gelfond-Schneider).

Eighth problem

See also: Assumption of Riemann

Démontrer the Assumption of Riemann.

In spite of the progress made by Sharp-edged in particular which showed the conjectures of Weil, and accepted for that the Médaille Fields in 1978, one is still far from to have solved this problem, which is announced like that of the 21e century.

Ninth problem

See also: Ninth problem of Hilbert

to Établir a law of reciprocity in the algebraic bodies of numbers.

An answer to this problem is brought by the law of reciprocity of Artin, shown by this one in 1927. This theorem enriches knowledge by the Théorie of the bodies of classes, whose development was facilitated by the introduction of the idèles by Chevalley in 1936.

Tenth problem

See also: Tenth problem of Hilbert

Trouver a algorithm determining if a equation diophantienne has solutions.

It was necessary to await work of Church and Turing in 1930 to define the concept of algorithm rigorously. In 1970, Yuri Matijasevic, establishing an equivalence between the units Recursively énumérable S and the units Diophantien S, established that such an algorithm could not not exist.

Eleventh problem

See also: Eleventh problem of Hilbert

Classifier quadratic forms with coefficients in the rings of entireties algébriques.

The Théorème of Hasse-Minkowski solves the problem on $\backslash \; mathbb\; Q$, and Siegel solved it on others just rings.

Twelfth problem

See also: Twelfth problem of Hilbert

Prolonger the theorem of Kronecker on the bodies non-abéliens.

Thirteenth problem

See also: Thirteenth problem of Hilbert

Montrer impossibility of solving the equation S of the seventh degree by means of functions from only two variables.

More generally, it is a question of studying the continuous functions (and, in fact, continuous functions of three variables) which cannot be expressed by composition starting from continuous functions of two variables. In 1954, Kolmogorov and its pupil Vladimir Arnold showed that this class was empty: there exists $n\; (2n+1)$ universal continuous functions $\backslash \; Phi\_\; \{ij\}$ (of $$ in $$) such as for any function continues $f:\; ^n\; \backslash \; to$, there exists $2n+1$ continuous functions $g\_j:\; \backslash \; to$ such as $f\; (x\_1,\; \backslash \; dowries,\; x\_n)\; =\; \backslash \; sum\_\; \{j=1\}\; ^\; \{2n+1\}\; g\_j\; \backslash \; left\; (\backslash \; sum\_\; \{i=1\}\; ^n\; \backslash \; Phi\_\; \{ij\}\; (x\_i)\; \backslash \; right)$. On the other hand, the question of the resolvability of the equation of the seventh degree by analytical functions of two variables is still open.

Fourteenth problem

See also: Fourteenth problem of Hilbert

Prouver finished character of certain complete systems of the fonctions.

The problem is the following: one considers a body $k$ and a subfield $K$ of $E\; =\; K\; (X\_1,\; \backslash \; dowries,\; X\_n)$; one poses $R\; =\; K,\; \backslash \; dowries,\; X\_n$; is the ring $K\; \backslash \; cup\; R$ a $k$-algèbre of the finished type? The answer is negative, like showed it Zariski (which gave following geometrical interpretation: there exists a projective Variété $X$ of body of the functions $K$ and an effective divider $D$ on $X$ such as $K\; \backslash \; cup\; R$ is the whole of the functions of $K$ having poles only on $R$). However, the search for conditions sufficient for the validity of the result of Hilbert caused very fertile idea in geometry. Nagata gave in 1959 a counterexample which showed the falseness of the conjecture.

Fifteenth problem

See also: Fifteenth problem of Hilbert

Mettre places from there the bases of the énumératif calculation of Hermann Schubert.

There it is a question of making rigorous certain calculations on the objects “in general position” in Théorie of the intersection, and in particular the “principle of conservation of the numbers”. This problem gave rise to the theories of the multiplicity of Samuel and Grothendieck.

Solved by Van der Waerden in 1930.

Sixteenth problem

See also: Sixteenth problem of Hilbert

Développer a Topology of the curved and algebraic surfaces.

This problem comprises two parts. The first relates to the number of real branches of an algebraic curve, and their provision; many modern results (Petrovskii, Thom, Arnold) bring information on their subject. The second part of the problem raises the question of the existence of a maximum number of cycles limiting for a linear differential equation defined by homogeneous polynomials of degree given; this question is still open.

Seventeenth problem

See also: Seventeenth problem of Hilbert

Montrer that a rational Fonction positive can be written in the form of nap of squares of functions rationnelles.

Solved by Artin in 1927. A purely logical demonstration was found by Robinson.

Eighteenth problem

See also: Eighteenth problem of Hilbert

Construire a Euclidean Space with Polyhedral S congruents.

The problem comprises three parts.

• Firstly, to show that there exists with isomorphism close only one finished number of discrete groups of isométries of $\backslash \; mathbb\; R^n$ admitting a compact fundamental field; this question was solved by Ludwig Bieberbach in 1910.
• Secondly, the question of the existence of polyhedrons which are not fundamental groups, but which can however pave space; such polyhedrons were built by Reinhardt and Heesch in the Thirties.
• Thirdly, this problem comprises also famous the Conjecture of Kepler about the stacking of the spheres in space, solved in 1998 by Thomas Hall.

Nineteenth problem

See also: Nineteenth problem of Hilbert

Prouver that the Calcul of the variations is always necessarily analytique.

Solved by Bernstein and Tibor Rado in 1929.

Twentieth problem

See also: Twentieth problem of Hilbert

Étudier the general solution of the problems of value limite.

Twenty-and-unième problem

See also: Twenty-and-unième problem of Hilbert

Prouver that any representation complexes of finished size can be obtained by action of monodromy on a differential equation of Fuchs.

Solved by Helmut Rörl in 1957.

Twenty-second problem

See also: Twenty-second problem of Hilbert

Uniformiser of the analytical curved by means of functions automorphes.

Solved by Koebe and Henri Poincaré in 1907.

Twenty-third problem

See also: Twenty-third problem of Hilbert

Développer a general method of resolution in the Calculation of the variations.

See too

• David Hilbert
• international Congress of mathematics
• Problems of the price of the millenium