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 differences between physiological and pathological states of parasympathetic ( and also of sympathetic ) dominance remain to be elucidated.

These are few and seemingly disjointed data, but they illustrate the important fact that fundamental alterations in conditioned reactions occur in a variety of states in which the hypothalamic balance has been altered by physiological experimentation, pharmacological action, or clinical processes.

Higher energy states are then similar to harmonics of the fundamental frequency.

Hierocles, writing in the 5th century, states that Ammonius ' fundamental doctrine was that Plato and Aristotle were in full agreement with each other:

Member states may, however, implement legislation which allows reopening of a case in the event that new evidence is found or if there was a fundamental defect in the previous proceedings.

The preamble of the charter provides that the members " reaffirm faith in fundamental human rights, in the equal rights of men and women " and Article 1 ( 3 ) of the United Nations charter states that one of the purposes of the UN is: " to achieve international cooperation in solving international problems of an economic, social, cultural, or humanitarian character, and in promoting and encouraging respect for human rights and for fundamental freedoms for all without distinction as to race, sex, language, or religion ".

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.

During the 17th century, the basic tenets of the Grotian or eclectic school, especially the doctrines of legal equality, territorial sovereignty, and independence of states, became the fundamental principles of the European political and legal system and were enshrined in the 1648 Peace of Westphalia.

The ILO Conventions which embody the fundamental principles have now been ratified by most member states.

Justin Meggitt states that there are fundamental similarities between the Josephus ' portrayal of John the Baptist and the New Testament narrative in that in both accounts John is positioned as a preacher of morality, not as someone who had challenged the political authority of Herod Antipas.

The idea of universal jurisdiction is fundamental to the operation of global organizations such as the United Nations and the International Court of Justice ( ICJ ), which jointly assert the benefit of maintaining legal entities with jurisdiction over a wide range of matters of significance to states ( the ICJ should not be confused with the ICC and this version of " universal jurisdiction " is not the same as that enacted in the War Crimes Law ( Belgium ) which is an assertion of extraterritorial jurisdiction that will fail to gain implementation in any other state under the standard provisions of public policy ).

Following a 1998 Arab League meeting in which fellow Arab states decided not to challenge U. N. sanctions, Gaddafi announced that he was turning his back on pan-Arab ideas, one of the fundamental tenets of his philosophy.

A fundamental concept of Lorentz's theory in 1895 was the " theorem of corresponding states " for terms of order v / c.

A European Union working party, for example, has announced a list of 44 recommendations to better harmonize, and if necessary pare back, the money laundering laws of EU member states to comply with fundamental privacy rights.

Only the " fundamental rights " under the federal constitution apply to Puerto Rico like the Privileges and Immunities Clause ( U. S. Constitution, Article IV, Section 2, Clause 1, also known as the ' Comity Clause ') that prevents a state from treating citizens of other states in a discriminatory manner, with regard to basic civil rights.

* Article 3 states that protection of the Antarctic environment as a wilderness with aesthetic and scientific value shall be a " fundamental consideration " of activities in the area.

In a classical system, a bit would have to be in one state or the other, but quantum mechanics allows the qubit to be in a superposition of both states at the same time, a property which is fundamental to quantum computing.

Modern physics attempts to explain every observed physical phenomenon by these fundamental interactions.

To every topological space X with distinguished point x < sub > 0 </ sub >, one can define the fundamental group based at x < sub > 0 </ sub >, denoted π < sub > 1 </ sub >( X, x < sub > 0 </ sub >).

Two ( or more ) inductors that have coupled magnetic flux form a transformer, which is a fundamental component of every electric utility power grid.

Rudimentary infantry skills such as basic individual movement techniques, shooting positions and field craft are fundamental to the training of every soldier.

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

Normal play Nim ( or more precisely the system of nimbers ) is fundamental to the Sprague-Grundy theorem, which essentially says that in normal play every impartial game is equivalent to a Nim heap that yields the same outcome when played in parallel with other normal play impartial games ( see disjunctive sum ).

If, however, the set of allowed candidates is expanded to the complex numbers, every non-constant polynomial has at least one root ; this is the fundamental theorem of algebra.

*" A fundamental rule of statutory construction requires that every part of a statute be presumed to have some effect, and not be treated as meaningless unless absolutely necessary.

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.

Poincaré's new condition — i. e., " trivial fundamental group "— can be restated as " every loop can be shrunk to a point.

With this statement he argues that rhetoric is a fundamental part of civic life in every society and that it has been necessary in the foundation of all aspects of society.

" Do the right thing " was a fundamental change in the user-interface model of previous news software ; rather than requiring users to navigate menus or learn a distinct command vocabulary for every operating mode of the program, certain single-keystroke commands were repeated throughout the user interface, performing the most obviously appropriate function for the task at hand.

Because one of the two fundamental postulates of special relativity says that the laws of physics should work the same way in every inertial frame, if it is possible for signals to move backwards in time in any one frame, it must be possible in all frames.

Broadly speaking, existentialists hold that there are certain fundamental questions that every human being must come to terms with if they are to take their subjective existences seriously and with intrinsic value.

The main theorem on covering spaces tells us that every subgroup H of G is the fundamental group of some covering space Y of X ; but every such Y is again a graph.

The sense of the expression damnatio memoriae and of the sanction is to cancel every trace of the person from the life of Rome, as if he had never existed, in order to preserve the honour of the city ; in a city that stressed the social appearance, respectability and the pride of being a true Roman as a fundamental requirement of the citizen, it was perhaps the most severe punishment.

Also, since every ontological primitive must be acknowledged as one of the fundamental principles of the natural world, we must also account for why this element in particular should be considered one of those underlying principles.

Richard Fuchs, author of An Unerring Fire, concludes, “ The affair at Fort Pillow was simply an orgy of death, a mass lynching to satisfy the basest of conduct – intentional murder – for the vilest of reasons – racism and personal enmity .” Andrew Ward downplays the controversy, “ Whether the massacre was premeditated or spontaneous does not address the more fundamental question of whether a massacre took place ... it certainly did, in every dictionary sense of the word .” John Cimprich states, “ The new paradigm in social attitudes and the fuller use of available evidence has favored a massacre interpretation ...

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root ( real coefficients and roots being within the definition of complex numbers ).

As will be mentioned again below, it follows from the fundamental theorem of algebra that every non-constant polynomial with real coefficients can be written as a product of polynomials with real coefficients whose degree is either 1 or 2.