The development of abstract algebra brought with itself group theory, rings and fields, Galois theory.

Ultimately, the abstract parallels between algebraic systems were seen to be more important than the details and modern algebra was born.

In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F, the ring of polynomials in the variable x with coefficients in F.

In the context of abstract algebra, for example, a mathematical object is an algebraic structure such as a group, ring, or vector space.

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry.

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.

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order ( the axiom of commutativity ).

The concept of an abelian group is one of the first concepts encountered in undergraduate abstract algebra, with many other basic objects, such as a module and a vector space, being its refinements.

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

* Alternative algebra, an abstract algebra with alternative multiplication

In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative.

Binary operations are the keystone of algebraic structures studied in abstract algebra: they form part of groups, monoids, semigroups, rings, and more.

In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice.

Homological algebra is category theory in its aspect of organising and suggesting manipulations in abstract algebra.

In abstract algebra, the derivative is interpreted as a morphism of modules of Kähler differentials.

It is also a tool used in branches of mathematics including combinatorics, abstract algebra, and mathematical analysis.

In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is invariant under all automorphisms of the parent group.

The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra.

In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.

A major driving force in the theoretical linguistic field is discovering the nature that language must have in the abstract in order to be learned in such a fashion.

In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible.

In abstract algebra, a finite field or Galois field ( so named in honor of Évariste Galois ) is a field that contains a finite number of elements.

In abstract algebra, field extensions are the main object of study in field theory.

Moreover, the original concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations, and the general field is often known as harmonic analysis.

In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry.

In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces ; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication.

Higher vocational education might be contrasted with education in a usually broader scientific field, which might concentrate on theory and abstract conceptual knowledge.

In each of these, a Reichskammer ( Reich Chamber ) was established, co-opting leading figures from the field ( usually not known Nazis ) to head each Chamber, and requiring them to supervise the purge of Jews, socialists and liberals, as well as practitioners of " degenerate " art forms such as abstract art and atonal music.

He was one of the best-known American Color field painters, although in the 1950s he was thought of as an abstract expressionist and in the early 1960s he was thought of as a minimalist painter.

Category theory, another field within " foundational mathematics ", is rooted on the abstract axiomatization of the definition of a " class of mathematical structures ", referred to as a " category ".

The continuation of abstract expressionism, color field painting, lyrical abstraction, geometric abstraction, minimalism, abstract illusionism, process art, pop art, postminimalism, and other late 20th-century modernist movements in both painting and sculpture continue through the first decade of the 21st century and constitute radical new directions in those mediums.

In abstract algebra, the field of fractions or field of quotients of an integral domain is the smallest field in which it can be embedded.

Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds, e. g., in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field: The set of all possible configurations of a system is modelled as a manifold, and this manifold's cotangent bundle describes the phase space of the system.

According to one text book: " On one plane the field is abstract, a set of analytical concepts about competition and monopoly.

The test of form is fidelity to the experience, a gauge also accepted by the abstract expressionist painters.

In the first instance, `` mimesis '' is here used to mean the recalling of experience in terms of vivid images rather than in terms of abstract ideas or conventional designations.

What I have observed time and time again is a process of integration, integration that begins as abstract design and gradually takes on recognizable form ; ;

for this division is but an abstract representation of the social struggle between mysticism and science.

An operational approach to sociology can never expect abstract certainty, since it is certainty which every new discovery in science either replaces or reshapes.

Its abstract decor is by John Hultberg.

In law, an abstract is a brief statement that contains the most important points of a long legal document or of several related legal papers.

The Abstract of Title, used in real estate transactions, is the more common form of abstract.

After this is accomplished, no abstract of title is necessary.

The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny quantities has been around for millennia, but these ideas were founded in abstract, philosophical reasoning rather than experimentation and empirical observation.

The most general setting in which these words have meaning is an abstract branch of mathematics called category theory.

This is a very abstract definition since, in category theory, morphisms aren't necessarily functions and objects aren't necessarily sets.

Walter Burkert notes that " Ares is apparently an ancient abstract noun meaning throng of battle, war.

He is also noted for his mastery of abstract approaches to mathematics and his perfectionism in matters of formulation and presentation.

The term is also used, especially in the description of algorithms, to mean associative array or " abstract array ", a theoretical computer science model ( an abstract data type or ADT ) intended to capture the essential properties of arrays.

(" the truth-values of our mathematical assertions depend on facts involving platonic entities that reside in a realm outside of space-time ") Whilst our knowledge of concrete, physical objects is based on our ability to perceive them, and therefore to causally interact with them, there is no parallel account of how mathematicians come to have knowledge of abstract objects.

Another line of defense is to maintain that abstract objects are relevant to mathematical reasoning in a way that is non causal, and not analogous to perception.