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Hilbert's and Nullstellensatz
* Hilbert's Nullstellensatz
It is even difficult to decide if a given algebraic system has complex solutions ( see Hilbert's Nullstellensatz ).
This relationship was discovered by David Hilbert who proved Nullstellensatz and several other important related theorems named after him ( like Hilbert's basis theorem ).
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.
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.
To see the connection with the classical picture, note that for any set S of polynomials ( over an algebraically closed field ), it follows from Hilbert's Nullstellensatz that the points of V ( S ) ( in the old sense ) are exactly the tuples ( a < sub > 1 </ sub >, ..., a < sub > n </ sub >) such that ( x < sub > 1 </ sub >-a < sub > 1 </ sub >, ..., x < sub > n </ sub >-a < sub > n </ sub >) contains S ; moreover, these are maximal ideals and by the " weak " Nullstellensatz, an ideal of any affine coordinate ring is maximal if and only if it is of this form.
For example, Hilbert's Nullstellensatz is a theorem which is fundamental for algebraic geometry, and is stated and proved in term of commutative algebra.
# REDIRECT Hilbert's Nullstellensatz

Hilbert's and German
* Hilbert's radio speech recorded in Königsberg 1930 ( in German ), with English translation
Hilbert's paradox of the Grand Hotel is a mathematical veridical paradox ( a non-contradictory speculation that is strongly counter-intuitive ) about infinite sets presented by German mathematician David Hilbert ( 1862 1943 ).
Hilbert's problems form a list of twenty-three problems in mathematics published by German mathematician David Hilbert in 1900.
The " 24th problem " ( in proof theory, on a criterion for simplicity and general methods ) was rediscovered in Hilbert's original manuscript notes by German historian Rüdiger Thiele in 2000.
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.

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

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

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