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, 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 field is a commutative ring which contains a multiplicative inverse for every nonzero element, equivalently a ring whose nonzero elements form an abelian group under multiplication.

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 extension of an object in abstract algebra, such as a group, is the underlying set of the object.

In abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of that field over which the polynomial splits or decomposes into linear factors.

In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian.

An abstract algebraic variety is a particular kind of scheme ; the generalization to schemes on the geometric side enables an extension of the correspondence described above to a wider class of rings.

In abstract algebra, the transcendence degree of a field extension L / K is a certain rather coarse measure of the " size " of the extension.

In mathematics, especially in the area of abstract algebra known as module theory, the injective hull ( or injective envelope ) of a module is both the smallest injective module containing it and the largest essential extension of it.

Further developments included: the extension by Spencer of the techniques to other structures of differential geometry ; the assimilation of the Kodaira-Spencer theory into the abstract algebraic geometry of Grothendieck, with a consequent substantive clarification of earlier work ; and deformation theory of other structures, such as algebras.

Concepts are abstract in that they omit the differences of the things in their extension, treating them as if they were identical.

The language used in the script varies between the naturalistic and the highly abstract or poetic, an extension of the style which Kane had developed in Crave, where she had begun significantly to marry form and content.

In abstract algebra, an algebraic field extension L / K is said to be normal if L is the splitting field of a family of polynomials in K. Bourbaki calls such an extension a quasi-Galois extension.

A Galois group G associated to a field extension L / K acts in a natural way on some abelian groups, for example those constructed directly from L, but also through other Galois representations that may be derived by more abstract means.

In mathematics, a pseudogroup is an extension of the group concept, but one that grew out of the geometric approach of Sophus Lie, rather than out of abstract algebra ( such as quasigroup, for example ).

The abstract syntactic relation of government in government and binding theory, a phrase structure grammar, is an extension of the traditional notion of case government.

In case of a so-called extensional functor we can in a sense abstract from the " material " part of its inputs and output, and regard the functor as a function turning directly the extension of its input ( s ) into the extension of its output.

Descartes axiomatically redefined the concept of matter in the Enlightenment to exclude any characteristics that would make it unsuitable for abstract, mathematical ( geometrical ) treatment: what has extension.

Jahrhundert ’, Zeitschrift für Musikwissenschaft, iv, 257 – 65 ), who employed it in reference to half of his perceived binarism between that 17th-century dance music which was in fact danced to, and that which was written in dance forms but was actually abstract music intended only for listening ( By extension, this duality applied to the dichotomy between any music for a specific purpose, Gebrauchsmusik, and that for none but the pleasure or edification of listening, Vortragsmusik ).