If we have a field extension F / K, which is to say a bigger field F that contains K, then there is a natural way to construct an algebra over F from any algebra over K. It is the same construction one uses to make a vector space over a bigger field, namely the tensor product.

We can now give it a central extension into the Lie algebra spanned by H ', P '< sub > i </ sub >, C '< sub > i </ sub >, L '< sub > ij </ sub > ( antisymmetric tensor ), M such that M commutes with everything ( i. e. lies in the center, that's why it's called a central extension ) and

The covariant derivative of a tensor field is presented as an extension of the same concept.

When the map φ between manifolds is a diffeomorphism, that is, it has a smooth inverse, then pullback can be defined for the vector fields as well as for 1-forms, and thus, by extension, for an arbitrary mixed tensor field on the manifold.

In mathematics, tensor calculus or tensor analysis is an advanced extension of vector calculus to more general mathematical objects called tensors.

This makes the extension from affine schemes, where it is just the tensor product of R-algebras, to all schemes of the fiber product operation a significant ( if technically anodyne ) result.

An extension of the fundamental theorem states that given a pseudo-Riemannian manifold there is a unique connection preserving the metric tensor with any given vector-valued 2-form as its torsion.

He was among the first to propose an extension of the gravitational theory based on the affine connection as the fundamental structure field rather than the metric tensor which was the original focus of general relativity.

** Every field extension has a transcendence basis.

In abstract algebra, a field extension L / K is called algebraic if every element of L is algebraic over K, i. e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i. e. which contain transcendental elements, are called transcendental.

For example, the field extension R / Q, that is the field of real numbers as an extension of the field of rational numbers, is transcendental, while the field extensions C / R and Q (√ 2 )/ Q are algebraic, where C is the field of complex numbers.

For instance, the field of all algebraic numbers is an infinite algebraic extension of the rational numbers.

If a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.

Every field has an algebraic extension which is algebraically closed ( called its algebraic closure ), but proving this in general requires some form of the axiom of choice.

An extension L / K is algebraic if and only if every sub K-algebra of L is a field.

In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.

The algebraic closure of a field K can be thought of as the largest algebraic extension of K.

Saying this another way, K is contained in a separably-closed algebraic extension field.

For example, if K is a field of characteristic p and if X is transcendental over K, is a non-separable algebraic field extension.

* Transcendence basis of a field extension

In the field of healthcare, funds were allocated towards modernisation and extension schemes aimed at improving administrative efficiency.

( To avoid any possible excuse for a dripping parade through your house, it is a good idea to have a telephone extension near the pool as well as a direct outdoor route between the pool, and the parking area.

An important extension of this idea is to consider the fundamental groupoid ( X, A ) where A is a set of " base points " and a subset of X.

Many such people believed in the idea that the United States is by definition benevolent, that the extension of its power, influence and hegemony is an extension of benevolence and brings freedom to those people subject to that hegemony.

* 1843 – Sir William Rowan Hamilton comes up with the idea of quaternions, a non-commutative extension of complex numbers.

Influential to thinkers associated with Postmodernism are Heidegger's critique of the subject-object or sense-knowledge division implicit in Rationalism, Empiricism and Methodological Naturalism, his repudiation of the idea that facts exist outside or separately from the process of thinking and speaking them ( however, Heidegger is not specifically a Nominalist ), his related admission that the possibilities of philosophical and scientific discourse are wrapped up in the practices and expectations of a society and that concepts and fundamental constructs are the expression of a lived, historical exercise rather than simple derivations of external, apriori conditions independent from historical mind and changing experience ( see Johann Gottlieb Fichte, Heinrich von Kleist, Weltanschauung and Social Constructionism ), and his Instrumentalist and Negativist notion that Being ( and, by extension, reality ) is an action, method, tendency, possibility and question rather than a discreet, positive, identifiable state, answer or entity ( see also Process Philosophy, Dynamism, Instrumentalism, Pragmatism and Vitalism ).

Infinite divisibility refers to the idea that extension, or quantity, when divided and further divided infinitely, cannot reach the point of zero quantity.

In any of several studies that treat the use of signs-for example, in linguistics, logic, mathematics, semantics, and semiotics-the extension of a concept, idea, or sign consists of the things to which it applies, in contrast with its comprehension or intension, which consists very roughly of the ideas, properties, or corresponding signs that are implied or suggested by the concept in question.

That a set can capture the notion of the extension of anything is the idea behind the axiom of extensionality in axiomatic set theory.

The extension of this idea to substances in general necessarily led him to the law of multiple proportions, and the comparison with experiment brilliantly confirmed his deduction.

Though strongly in favour of the extension of individual legal rights, he opposed the idea of natural law and natural rights, calling them " nonsense upon stilts ".

The BIOS on the IBM-PC class machines was an extension of this idea and has accreted more features and functions in the 20 years since the first IBM-PC was introduced in 1981.

This letter also made it clear that an extension of the proposed colony to include a penal settlement at the Chatham Islands was envisaged: ‘ The King is still preoccupied with the idea and with the necessity of a place of deportation.

By extension, this may have later led to the idea of "' the son of man '," an eschatological Messianic figure, within Judaism.

A few Lutheran churches, such as the Church of Sweden, allows for the formation of political parties ( also known as nominating groups ()) to nominate candidates for the Synod ; it is an extension of the idea of multi-ideological democracy within the church.

By extension from its primary sense, the idea that a virgin has a sexual " blank slate ", unchanged by any past intimate connection or experience, leads to the abstraction of unadulterated purity.

An extension of this idea is that the human mind is itself a computational system, and hence providing it with raw data in as effective a way as possible is crucial to research.

Indeed, some recent and controversial works, such as Huntington's The Clash of Civilizations and the Remaking of World Order can be regarded as an extension of comparative study into the idea of conflicts among the groups compared.

Out of his sight, the three eavesdrop using an extension phone while Shields describes his new idea, and become more and more interested.

Ritzer ’ s idea of McDonaldization is an extension of Max Weber ’ s ( 1864 – 1920 ) classical theory of the rationalization of modern society and culture.

For the steam ferry Etna, which entered service at Tranmere on 17 April 1817, the idea of extension stages was mooted.

This brought together thinkers with interests in artificial intelligence, nanotechnology, genetic engineering, life extension, mind uploading, idea futures, robotics, space exploration, memetics, and the politics and economics of transhumanism.

In the Tenth Elucidation, for instance, Malebranche introduced his theory of " intelligible extension ", a single, archetypal idea of extension into which the ideas of all particular kinds of bodies could be jointly resolved.

With the death of God, one must also accept by extension the idea of universal guilt and the impossibility of innocence.