* Krull's theorem ( 1929 ): Every ring with a multiplicative identity has a maximal ideal.

Krull's principal ideal theorem states that every principal ideal in a commutative Noetherian ring has height one ; that is, every principal ideal is contained in a prime ideal minimal amongst nonzero prime ideals.

The proof of the general case requires some commutative algebra, namely the fact, that the Krull dimension of k is one — see Krull's principal ideal theorem ).

In a Noetherian ring, Krull's height theorem says that the height of an ideal generated by n elements is no greater than n.

# REDIRECT Krull's principal ideal theorem

In commutative algebra, Krull's principal ideal theorem, named after Wolfgang Krull ( 1899 – 1971 ), gives a bound on the height of a principal ideal in a Noetherian ring.

This theorem can be generalized to ideals that are not principal, and the result is often called Krull's height theorem.

# REDIRECT Krull's principal ideal theorem

** Krull's intersection theorem

** Krull's principal ideal theorem

** Krull's theorem

* Krull's intersection theorem

* Krull's principal ideal theorem

* Krull's theorem

Then its coefficients generate a proper ideal I, which by Krull's theorem ( which depends on the axiom of choice ) is contained in a maximal ideal m of R. Then R / m is a field, and ( R / m ) is therefore an integral domain.

** Well-ordering theorem: Every set can be well-ordered.

Note that " completeness " has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement such that neither nor can be proved from the given set of axioms.

The notion of the Kolmogorov complexity can be used to state and prove impossibility results akin to Gödel's incompleteness theorem and Turing's halting problem.

which can be viewed as a version of the Pythagorean theorem.

For a first order predicate calculus, with no (" proper ") axioms, Gödel's completeness theorem states that the theorems ( provable statements ) are exactly the logically valid well-formed formulas, so identifying valid formulas is recursively enumerable: given unbounded resources, any valid formula can eventually be proven.

* Metamath-a language for developing strictly formalized mathematical definitions and proofs accompanied by a proof checker for this language and a growing database of thousands of proved theorems ; while the Metamath language is not accompanied with an automated theorem prover, it can be regarded as important because the formal language behind it allows development of such a software ; as of March, 2012, there is no " widely " known such software, so it is not a subject of " automated theorem proving " ( it can become such a subject ), but it is a proof assistant.

Antiderivatives are important because they can be used to compute definite integrals, using the fundamental theorem of calculus: if F is an antiderivative of the integrable function f, then:

The fundamental theorem of arithmetic states that any positive integer n can be represented uniquely as a product of powers of primes: where p < sub > 1 </ sub > < p < sub > 2 </ sub > < ... < p < sub > k </ sub > are primes and the a < sub > j </ sub > are positive integers.

Using the above theorem it is easy to see that the original Borsuk Ulam statement is correct since if we take a map f: S < sup > n </ sup > → ℝ < sup > n </ sup > that does not equalize on any antipodes then we can construct a map g: S < sup > n </ sup > → S < sup > n-1 </ sup > by the formula

* The Ham sandwich theorem: For any compact sets A < sub > 1 </ sub >,..., A < sub > n </ sub > in ℝ < sup > n </ sup > we can always find a hyperplane dividing each of them into two subsets of equal measure.

* The case n = 1 can be illustrated by the claim that there always exist a pair of opposite points on the earth's equator with the same temperature, this can be shown to be true much more easily using the intermediate value theorem.

The binomial theorem can be applied to the powers of any binomial.

For a binomial involving subtraction, the theorem can be applied as long as the opposite of the second term is used.

For positive values of a and b, the binomial theorem with n = 2 is the geometrically evident fact that a square of side can be cut into a square of side a, a square of side b, and two rectangles with sides a and b. With n = 3, the theorem states that a cube of side can be cut into a cube of side a, a cube of side b, three a × a × b rectangular boxes, and three a × b × b rectangular boxes.

A class of algorithms called SAT solvers can efficiently solve a large enough subset of SAT instances to be useful in various practical areas such as circuit design and automatic theorem proving, by solving SAT instances made by transforming problems that arise in those areas.

Schaefer's dichotomy theorem states that, for any restriction to Boolean operators that can be used to form these subformulae, the corresponding satisfiability problem is in P or NP-complete.

In his 1799 doctorate in absentia, A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

A number of scholars claim that Gödel's incompleteness theorem proves that any attempt to construct a ToE is bound to fail.

This means that if we try to use the implicit function theorem to express y as a function of x near y = 0, we shall fail ; and indeed no linear combination of x and y is a function of another essentially different one, so that this is a geometric condition not tied to any choice of coordinate axes.

If the field K is not algebraically closed, the theorem can fail.

For algebraically closed fields of characteristic p > 0 Lie's theorem holds provided the dimension of the representation is less than p, but can fail for representations of dimension p. An example is given by the 3-dimensional nilpotent Lie algebra spanned by 1, x, and d / dx acting on the p-dimensional vector space k /( x < sup > p </ sup >), which has no eigenvectors.

With weaker conditions, uniqueness can fail, according to the Sonnenschein – Mantel – Debreu theorem.

First, the strong completeness theorem and compactness fail.

In Verified Design by Contract, the contracts are verified by static analysis and automated theorem proving, so that it is certain that they will not fail at runtime.

In the primary decomposition theorem, it is not necessary that the vector space V be finite dimensional, nor is it necessary for parts ( A ) and ( B ) that P be the minimal polynomial for T.

One motivation for this use is that a number of generally accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs.

The debate is interesting enough, however, that it is considered of note when a theorem in ZFC ( ZF plus AC ) is logically equivalent ( with just the ZF axioms ) to the axiom of choice, and mathematicians look for results that require the axiom of choice to be false, though this type of deduction is less common than the type which requires the axiom of choice to be true.

Assuming ZF is consistent, Paul Cohen employed the technique of forcing, developed for this purpose, to show that the axiom of choice itself is not a theorem of ZF by constructing a much more complex model which satisfies ZF ¬ C ( ZF with the negation of AC added as axiom ) and thus showing that ZF ¬ C is consistent.

** Stone's representation theorem for Boolean algebras needs the Boolean prime ideal theorem.

** Gödel's completeness theorem for first-order logic: every consistent set of first-order sentences has a completion.

The original Nernst heat theorem makes the weaker and less controversial claim that the entropy change for any isothermal process approaches zero as T → 0:

This explains the failure of the classical equipartition theorem for metals that eluded classical physicists in the late 19th century.

Mordell's theorem had an ad hoc proof ; Weil began the separation of the infinite descent argument into two types of structural approach, by means of height functions for sizing rational points, and by means of Galois cohomology, which was not to be clearly named as that for two more decades.

The " heuristic " approach of the Logic Theory Machine tried to emulate human mathematicians, and could not guarantee that a proof could be found for every valid theorem even in principle.

Bandwidth typically refers to baseband bandwidth in the context of, for example, sampling theorem and Nyquist sampling rate, while it refers to passband bandwidth in the context of Nyquist symbol rate or Shannon-Hartley channel capacity for communication systems.

The binomial theorem also holds for two commuting elements of a Banach algebra.

Given x ∈ A, the holomorphic functional calculus allows to define ƒ ( x ) ∈ A for any function ƒ holomorphic in a neighborhood of Furthermore, the spectral mapping theorem holds:

Greek mathematician Euclid mentioned the special case of the binomial theorem for exponent 2 as did the 3rd century B. C.

The most basic example of the binomial theorem is the formula for the square of x + y:

If one integrates this picture, which corresponds to applying the fundamental theorem of calculus, one obtains Cavalieri's quadrature formula, the integral – see proof of Cavalieri's quadrature formula for details.