Hilbert's tenth problem was to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution.

The is related to Hilbert's tenth problem, which asks for an algorithm to decide whether Diophantine equations have a solution.

# REDIRECT Hilbert's tenth problem

Matiyasevich's completion of the MRDP theorem settled Hilbert's tenth problem.

Hilbert's tenth problem was to find a general algorithm which can decide whether a given Diophantine equation has a solution among the integers.

While Hilbert's tenth problem is not a formal mathematical statement as such the nearly universal acceptance of the ( philosophical ) identification of a decision algorithm with a total computable predicate allows us to use the MRDP theorem to conclude the tenth problem is unsolvable.

Hilbert's tenth problem is the tenth on the list of Hilbert's problems of 1900.

The unsolvability of Hilbert's tenth problem is a consequence of the surprising fact that the

Because there exists a recursively enumerable set that is not computable, the unsolvability of Hilbert's tenth problem is an immediate consequence.

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.

In 1970, Yuri Matiyasevich proved ( using results of Julia Robinson ) Matiyasevich's theorem, which implies that Hilbert's tenth problem has no effective solution ; this problem asked whether there is an effective procedure to decide whether a Diophantine equation over the integers has a solution in the integers.

1970: Hilbert's tenth problem is proven unsolvable: there is no recursive solution to decide whether a Diophantine equation ( multivariable polynomial equation ) has a solution in integers.

He is best known for his negative solution of Hilbert's tenth problem, presented in his doctoral thesis, at LOMI ( the Leningrad Department of the Steklov Institute of Mathematics ).

In an account that had become standard by the mid-century, Hilbert's problem set was also a kind of manifesto, that opened the way for the development of the formalist school, one of three major schools of mathematics of the 20th century.

In logic, the second problem on David Hilbert's list of open problems presented in 1900 was to prove that the axioms of arithmetic are consistent.

* Hilbert's second problem

# REDIRECT Hilbert's fifth problem

Hilbert's fifth problem asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples.

He also published papers on mathematical logic, and solved a special case of Hilbert's fifth problem.

( See Hilbert's second problem.

Work on Hilbert's 10th problem led in the late twentieth century to the construction of specific Diophantine equations for which it is undecidable whether they have a solution, or even if they do, whether they have a finite or infinite number of solutions.

More fundamentally, Hilbert's first problem was on the continuum hypothesis.

In 1900, David Hilbert posed an influential question about transcendental numbers, Hilbert's seventh problem: If a is an algebraic number, that is not zero or one, and b is an irrational algebraic number, is a < sup > b </ sup > necessarily transcendental?

In Hilbert's sixth problem, he challenged researchers to find an axiomatic basis to all of physics.

Church and Turing independently demonstrated that Hilbert's Entscheidungsproblem ( decision problem ) was unsolvable, thus identifying the computational core of the incompleteness theorem.

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, topology, algebraic geometry, classical mechanics and singularity theory, including posing the ADE classification problem, since his first main result — the partial solution of Hilbert's thirteenth problem in 1957 at the age of 19.

While a student of Andrey Kolmogorov at Moscow State University and still a teenager, Arnold showed in 1957 that any continuous function of several variables can be constructed with a finite number of two-variable functions, thereby partially solving Hilbert's thirteenth problem.

In fact, Smale's list contains some of the original Hilbert problems, including the Riemann hypothesis and the second half of Hilbert's sixteenth problem, both of which are still unsolved.

The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to Hilbert's second problem.

Many logicians believe that Gödel's incompleteness theorems struck a fatal blow to David Hilbert's second problem, which asked for a finitary consistency proof for mathematics.

Not all mathematicians agree with this analysis, however, and the status of Hilbert's second problem is not yet decided ( see " Modern viewpoints on the status of the problem ").

* Hilbert's second 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.

Zariski's formulation of Hilbert's fourteenth problem asks whether, for a quasi-affine algebraic variety X over a field k, possibly assuming X normal or smooth, the ring of regular functions on X is finitely generated over k.

Hilbert's eleventh problem asks for a similar theory.

One interpretation of Hilbert's twelfth problem asks to provide a suitable analogue of exponential, elliptic, or modular functions, whose special values would generate the maximal abelian extension K < sup > ab </ sup > of a general number field K. In this form, it remains unsolved.

However the construction of K < sup > ab </ sup > in class field theory involves first constructing larger non-abelian extensions using Kummer theory, and then cutting down to the abelian extensions, so does not really solve Hilbert's problem which asks for a more direct construction of the abelian extensions.

Hilbert's problem asks whether the minimizers w of an energy functional such as

One of the important problems for logicians in the 1930s was David Hilbert's Entscheidungsproblem, which asked if there was a mechanical procedure for separating mathematical truths from mathematical falsehoods.

As a result, he demonstrated Hilbert's basis theorem: showing the existence of a finite set of generators, for the invariants of quantics in any number of variables, but in an abstract form.

Gordan, the house expert on the theory of invariants for the Mathematische Annalen, was not able to appreciate the revolutionary nature of Hilbert's theorem and rejected the article, criticizing the exposition because it was insufficiently comprehensive.

In fact, Minkowski seems responsible for most of Hilbert's physics investigations prior to 1912, including their joint seminar in the subject in 1905.

* Tarski's axioms: Alfred Tarski ( 1902 – 1983 ) and his students defined elementary Euclidean geometry as the geometry that can be expressed in first-order logic and does not depend on set theory for its logical basis, in contrast to Hilbert's axioms, which involve point sets.

For instance, the ring of integers and the polynomial ring over a field are both Noetherian rings, and consequently, such theorems as the Lasker – Noether theorem, the Krull intersection theorem, and the Hilbert's basis theorem hold for them.

This fact follows from the famous Hilbert's basis theorem named after mathematician David Hilbert ; the theorem asserts that if R is any Noetherian ring ( such as, for instance, ), R is also a Noetherian ring.

Hilbert's system consisting of 20 axioms < ref > a 21 < sup > st </ sup > axiom appeared in the French translation of Hilbert's Grundlagen der Geometrie according to most closely follows the approach of Euclid and provides the justification for all of Euclid's proofs.

The early development of mathematical proof theory was driven by the desire to provide finitary consistency proofs for all of mathematics as part of Hilbert's program.

Hilbert's Nullstellensatz ( German for " theorem of zeros ," or more literally, " zero-locus-theorem " – see Satz ) is a theorem which establishes a fundamental relationship between geometry and algebra.

0 for all f in I. Hilbert's Nullstellensatz states that if p is some polynomial in kX < sub > n </ sub > which vanishes on the algebraic set V ( I ), i. e. p ( x ) = 0 for all x in V ( I ), then there exists a natural number r such that p < sup > r </ sup > is in I.

It is now common to interpret Hilbert's second question as asking in particular for a proof that Peano arithmetic is consistent.

This theorem shows that if the only acceptable proof procedures are those that can be formalized within arithmetic then Hilbert's call for a consistency proof cannot be answered.