has no zero in F. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed.

Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.

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 algebra states that the algebraic closure of the field of real numbers is the field of complex numbers.

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.

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.

It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus.

The corresponding form of the fundamental theorem of calculus is Stokes ' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve.

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 fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the set of real numbers and the set of natural numbers do not have the same cardinal number.

The fundamental theorem of calculus states that antidifferentiation is the same as integration.

It considers the connection between perfect numbers and Mersenne primes, the infinitude of prime numbers, Euclid's lemma on factorization ( which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations ), and the Euclidean algorithm for finding the greatest common divisor of two numbers.

Furthermore, Bob is only able to perform his measurement once: there is a fundamental property of quantum mechanics, known as the " no cloning theorem ", which makes it impossible for him to make a million copies of the electron he receives, perform a spin measurement on each, and look at the statistical distribution of the results.

Liouville's theorem may be used to elegantly prove the fundamental theorem of algebra.

Finally, it is a basic tool for proving theorems in modern number theory, such as Lagrange's four-square theorem and the fundamental theorem of arithmetic ( unique factorization ).

In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism.

In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 is either prime itself or is the product of prime numbers, and that, although the order of the primes in the second case is arbitrary, the primes themselves are not.

The fundamental components of a general-purpose computer are arithmetic logic unit | arithmetic and logic unit, control unit | control circuitry, storage space, and input / output devices.

As the name indicates, the fundamental theorem of arithmetic is true in them.

The fundamental theorem of arithmetic is encoded by the von Mangoldt function.

* PlanetMath: Proof of fundamental theorem of arithmetic

By the fundamental theorem of arithmetic, every positive integer has a unique prime factorization.

) However, the fundamental theorem of arithmetic gives no insight into how to obtain an integer's prime factorization ; it only guarantees its existence.

The fundamental theorem of arithmetic guarantees that there is only one possible string that will be accepted ( providing the factors are required to be listed in order ), which shows that the problem is in both UP and co-UP.

A multiplicative function is completely determined by its values at the powers of prime numbers, a consequence of the fundamental theorem of arithmetic.

The fundamental theorem of arithmetic continues to hold in unique factorization domains.

The prime ideals of the ring of integers are the ideals ( 0 ), ( 2 ), ( 3 ), ( 5 ), ( 7 ), ( 11 ), … The fundamental theorem of arithmetic generalizes to the Lasker – Noether theorem, which expresses every ideal in a Noetherian commutative ring as an intersection of primary ideals, which are the appropriate generalizations of prime powers.

* It purports to reveal the fundamental basis for arithmetic.

Principal ideal domains are thus mathematical objects which behave somewhat like the integers, with respect to divisibility: any element of a PID has a unique decomposition into prime elements ( so an analogue of the fundamental theorem of arithmetic holds ); any two elements of a PID have a greatest common divisor ( although it may not be possible to find it using the Euclidean algorithm ).

An important example is the ring Z / nZ of integers modulo n. If n is written as a product of prime powers ( see fundamental theorem of arithmetic ):

The proof of Euler's identity uses only the formula for the geometric series and the fundamental theorem of arithmetic.

In addition to the fundamental operations of arithmetic and mathematical functions, modern spreadsheets provide built-in functions for common financial and statistical operations.

Now is an integer, and so can be factored as a product of prime numbers, by the fundamental theorem of arithmetic.

This is a consequence of the fundamental theorem of arithmetic.

The fundamental theorem of arithmetic states that if n > 1 there is a unique expression for n,

If n is non-zero, and is neither 1 nor − 1, by the fundamental theorem of arithmetic, there exist primes p < sub > i </ sub >, and positive integers e < sub > i </ sub >, with.

The internal workings of the FSM are governed by the 1979 constitution, which guarantees fundamental human rights and establishes a separation of governmental powers.

Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic.

The most fundamental results of this theory are Shannon's source coding theorem, which establishes that, on average, the number of bits needed to represent the result of an uncertain event is given by its entropy ; and Shannon's noisy-channel coding theorem, which states that reliable communication is possible over noisy channels provided that the rate of communication is below a certain threshold, called the channel capacity.

The constitution promulgated at independence on 12 July 1979, establishes the Republic of Kiribati as a sovereign democratic republic and guarantees the fundamental rights of its citizens.

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.

The religious beliefs of the citizenry, even though they include fundamental principles of Christianity such as the immortality of the soul and divine providence, form a natural religion that establishes a sort of osmosis between the city and the stars.

Akulov ( in 1972 ) and J. Wess and B. Zumino ( in 1974 ) independently rediscovered supersymmetry, a radically new type of symmetry of spacetime and fundamental fields, which establishes a relationship between elementary particles of different quantum nature, bosons and fermions, and unifies spacetime and internal symmetries of the microscopic world.

It lays down the framework defining fundamental political principles, establishes the structure, procedures, powers, and duties of government institutions, and sets out fundamental rights, directive principles, and the duties of citizens.

Thus, their theory establishes fundamental connections between graph minors and topological embeddings of graphs.

It remains the current fundamental law that establishes the structure of the government of Romania, the rights and obligations of the country's citizens, and its mode of passing laws.

If the court establishes that the right being violated is a fundamental right, it applies strict scrutiny.

In single-variable calculus, the fundamental theorem of calculus establishes a link between the derivative and the integral.

If the court establishes that the right being violated is a fundamental right, it applies strict scrutiny.

Cell culture is a fundamental component of tissue culture and tissue engineering, as it establishes the basics of growing and maintaining cells ex vivo.

It establishes the framework and principles of government, and lists the fundamental rights and duties of Chinese citizens.

DLTS establishes fundamental defect parameters and measures their concentration in the material.

According to Sterba, the liberty of the poor should be morally prioritized in light of the fundamental moral principle ‘ “ Ought ” implies “ Can ” ’ from which it follows that it would be unreasonable to ask the poor to relinquish their liberty not be interfered with, noting that “ in the extreme case it would involve asking or requiring the poor to sit back and starve to death ” and that “ by contrast it would not be unreasonable to ask and require the rich to sacrifice their liberty to meet some of their needs so that the poor can have the liberty to meet their basic needs .” Having argued that ‘ ” Ought ” implies “ Can ” ’ establishes the reasonability of asking the rich to sacrifice their luxuries for the basic needs of the poor, Sterba invokes a second fundamental principle, “ The Conflict Resolution Principle ” to argue that it is reasonable to make it a moral requirement.

: Von Staudt made the important discovery that the relation which a conic establishes between poles and polars is really more fundamental than the conic itself, and can be set up independently.