** 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 )

* Hilbert's syzygy theorem

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.

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.

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.

Hilbert's syzygy theorem then states that there exists a free resolution of M of length at most n.

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.

However, in Hilbert's aptly named Grand Hotel, the quantity of odd-numbered rooms is no smaller than total " number " of rooms.

Hilbert's geometry is mathematical, because it talks about abstract points, but in Field's theory, these points are the concrete points of physical space, so no special mathematical objects at all are needed.

While the theorems of Gödel and Gentzen are now well understood by the mathematical logic community, no consensus has formed on whether ( or in what way ) these theorems answer Hilbert's second problem.

The second part of Hilbert's eighteenth problem asked for a single polyhedron tiling Euclidean 3-space, such that no tiling by it is isohedral ( an anisohedral tile ).

However, Gödel's incompleteness theorems showed in 1931 that no finite system of axioms, if complex enough to express our usual arithmetic, could ever fulfill the goals of Hilbert's program, demonstrating many of Hilbert's aims impossible, and specifying limits on most axiomatic systems.

Unlike some other modern axiomatizations, such as Birkhoff's and Hilbert's, Tarski's axiomatization has no primitive objects other than points, so a variable or constant cannot refer to a line or an angle.