* Hilbert's syzygy theorem

In mathematics, Hilbert's syzygy theorem is a result of commutative algebra, first proved by David Hilbert ( 1890 ) in connection with the syzygy ( relation ) problem of invariant theory.

# REDIRECT Hilbert's syzygy theorem

Let A = K be the ring of polynomials in n variables over a field K. Then the global dimension of A is equal to n. This statement goes back to David Hilbert's foundational work on homological properties of polynomial rings, see Hilbert's syzygy theorem.

** Hilbert's basis theorem

Hilbert's first work on invariant functions led him to the demonstration in 1888 of his famous finiteness theorem.

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 1931 his incompleteness theorem showed that Hilbert's grand plan was impossible as stated.

* Hilbert's basis theorem

* Hilbert's irreducibility theorem

* Hilbert's theorem ( differential geometry )

The process of abstract axiomatization as exemplified by Hilbert's axioms reduces geometry to theorem proving or predicate logic.

In mathematics, specifically commutative algebra, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated.

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.

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.

By induction, Hilbert's basis theorem establishes that, the ring of all polynomials in n variables with coefficients in, is a Noetherian ring.

For a proof of this result, see the corresponding section on the Hilbert's basis theorem page.

# REDIRECT Hilbert's basis theorem

In Hilbert's axiomatization of geometry this statement is given as a theorem, but only after much groundwork.

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

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 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.

In 1920 he proposed explicitly a research project ( in metamathematics, as it was then termed ) that became known as Hilbert's program.

Hilbert's work had started logic on this course of clarification ; the need to understand Gödel's work then led to the development of recursion theory and then mathematical logic as an autonomous discipline in the 1930s.

He started with the " betweenness " of Hilbert's axioms to characterize space without coordinatizing it, and then added extra relations between points to do the work formerly done by vector fields.

* Richard Zach, 2006, " Hilbert's program then and now ", in Philosophy of Logic, Dale Jacquette ( ed.

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.

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.

Kurt Gödel's seminal work on proof theory first advanced, then refuted this program: his completeness theorem initially seemed to bode well for Hilbert's aim of reducing all mathematics to a finitist formal system ; then his incompleteness theorems showed that this is unattainable.

1 and n = 2 by Zariski in 1954 ) then in 1959 Masayoshi Nagata found a counterexample to Hilbert's conjecture.

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.

Arnold then expanded on this work to show that only two-variable functions were in fact required, thus answering Hilbert's question.

Hilbert's goals of creating a system of mathematics that is both complete and consistent were dealt a fatal blow by the second of Gödel's incompleteness theorems, which states that sufficiently expressive consistent axiom systems can never prove their own consistency.

* Hilbert's theorem ( 1901 ) states that there exists no complete analytic ( class C < sup > ω </ sup >) regular surface in R < sup > 3 </ sup > of constant negative Gaussian curvature.

In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert, states that every finite number of irreducible polynomials in a finite number of variables and having rational number coefficients admit a common specialization of a proper subset of the variables to rational numbers such that all the polynomials remain irreducible.

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.

Hilbert's example: " the assertion that either there are only finitely many prime numbers or there are infinitely many " ( quoted in Davis 2000: 97 ); and Brouwer's: " Every mathematical species is either finite or infinite.

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

However, the question is still debated since in the literature there have been other such claims, largely based on different interpretations of Hilbert's statement of the problem given by various researchers.

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.

We have already noted the example of analytic geometry, and more generally the field of algebraic geometry thoroughly develops the connections between geometric objects ( algebraic varieties, or more generally schemes ) and algebraic ones ( ideals ); the touchstone result here is Hilbert's Nullstellensatz which roughly speaking shows that there is a natural one-to-one correspondence between the two types of objects.

Courant eventually became David Hilbert's assistant in Göttingen and obtained his doctorate there in 1910.

Namely, there exist topological manifolds which admit no C < sup > 1 </ sup >− structure, a result proved by, and later explained in the context of Donaldson's theorem ( compare Hilbert's fifth problem ).

The work of Lubotzky and Mann, combined with Michel Lazard's solution to Hilbert's fifth problem over the p-adic numbers, shows that a pro-p group is p-adic analytic if and only if it has finite rank, i. e. there exists a positive integer such that any closed subgroup has a topological generating set with no more than elements.