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.

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 establishes the central role of primes in number theory: any integer greater than 1 can be expressed as a product of primes that is unique up to ordering.

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 constitution provides for a parliamentary system of government and guarantees certain fundamental rights.

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

The most important of these are article 19, which generally requires a statutory basis for any infringements of the fundamental rights guaranteed by the Constitution while also guaranteeing judicial review ; article 20, paragraph 3, which guarantees the rule of law ; and article 3 which guarantees equality.

The plan provided specific guarantees of fundamental human rights.

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.

The constitution provides for a parliamentary system of government and guarantees certain fundamental rights.

The fundamental impetus for this political transformation was a recommendation by the Council of Europe in 1990 that, if Andorra wished to attain full integration in the European Union ( EU ), it should adopt a modern constitution that guarantees the rights of those living and working there.

It is a best-effort network without fundamental Quality of Service ( QoS ) guarantees.

According to Greene and Armstrong, some strategists assert adhering to the fundamental principles guarantees victory, while others claim war is unpredictable and the general must be flexible in formulating a strategy.

Trudeau spoke on television in October 1980, and announced his intention to constitutionalize a bill of rights that would include fundamental freedoms, democratic guarantees, freedom of movement, legal rights, equality and language rights.

It guarantees certain fundamental rights, along with a popularly elected non-executive president, a bicameral parliament based on the Westminster system, a separation of powers and judicial review.

The Constitution established a parliamentary system of government under a form of constitutional monarchy, and contained guarantees of certain fundamental rights.

The European Convention on Human Rights defines and guarantees since 1950 human rights and fundamental freedoms in Europe.

Current strategies for improvement in pain management include framing it as an ethical issue ; promoting pain management as a legal right ; providing constitutional guarantees and statutory regulations that span negligence law, criminal law, and elder abuse ; defining pain management as a fundamental human right ; categorizing failure to provide pain management as professional misconduct, and issuing guidelines and standards of practice by professional bodies.

Section 22 guarantees the most fundamental right of all-the right to life.

Supporters lauded the constitution's guarantees of " fundamental rights ":

SICA has the fundamental aim to realize an integration that is political and ideological representative in a pluralistic democratic system that guarantees free elections and participation on equal conditions for political parties.

The role of the incorporation doctrine in applying the guarantees of the Bill of Rights to the states is just as notable as the use of due process to define new fundamental rights that are not explicitly guaranteed by the Constitution's text.

He subscribed to the doctrine that the clause not only provided procedural guarantees, but also protected a wide range of fundamental rights, including those that were not specifically mentioned in the text of the Constitution.

In this regard, the constitution or other fundamental law of the new State or States shall include specific guarantees respecting -

The application of DSM-IV-TR criteria to the various pieces of criminal as well as federal health and provincial mental health law in Canada seems still to occur in spite of their conflict in this respect with Section 2 ( b ) of the Canadian Charter of Rights and Freedoms, which guarantees the " fundamental " " freedom of thought, belief, opinion and expression ".

The Republic of Turkey has entered various human rights commitments, some of which are expressed in the 1982 Turkish Constitution, Part Two of which guarantees " fundamental rights and freedoms " such as the right to life, security of person, and right to property.