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.

The term abstract data type can also be regarded as a generalised approach of a number of algebraic structures, such as lattices, groups, and rings.

This point of view is useful in abstract algebraic dimension theory due to Gabriel and Rentschler.

To abstract from the number of inputs, outputs and states, the variables are expressed as vectors and the differential and algebraic equations are written in matrix form ( the latter only being possible when the dynamical system is linear ).

In mathematics and abstract algebra, a group is the algebraic structure, where is a non-empty set and denotes a binary operation called the group operation.

This led to modern abstract algebraic notions such as Euclidean domains.

In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures ( such as groups, rings, or vector spaces ).

Thus one can understand equations by a pure understanding of abstract topology or geometry — this idea is of importance in algebraic geometry.

Similarly, a mathematician does not restrict his study of numbers to the integers ; rather he considers more abstract structures such as rings, and in particular number rings in the context of algebraic number theory.

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element.

In advanced mathematics, polynomials are used to construct polynomial rings, a central concept in abstract algebra and algebraic geometry.

Unlike Structuralists, however, the Post-structuralists questioned the division between relation and component and, correspondingly, did not attempt to reduce the subjects of their study to an essential set of relations that could be portrayed with abstract, functional schemes or mathematical symbols ( as in Claude Lévi-Strauss's algebraic formulation of mythological transformation in " The Structural Study of Myth ").

In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that " division " is always possible.

Semigroups are important in many areas of mathematics because they are the abstract algebraic underpinning of " memoryless " systems: time-dependent systems that start from scratch at each iteration.

While this definition is the most abstract, it is also the one most easily transferred to other settings, for instance to the varieties considered in algebraic geometry.

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.

The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and axioms.

The process of abstract axiomatization as exemplified by Hilbert's axioms reduces geometry to theorem proving or predicate logic.

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.

Mathematicians do research in fields such as logic, set theory, category theory, abstract algebra, number theory, analysis, geometry, topology, dynamical systems, combinatorics, game theory, information theory, numerical analysis, optimization, computation, probability and statistics.

Other examples of mathematical objects might include lines and planes in geometry, or elements and operations in abstract algebra.

Hilbert's geometry is mathematical, because it talks about abstract points, but in Field's theory, these points are the concrete points of physical space, so no special mathematical objects at all are needed.

Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory and abstract algebra.

The discovery of a consistent alternative geometry that might correspond to the structure of the universe helped to free mathematicians to study abstract concepts irrespective of any possible connection with the physical world.

( One should keep in mind that in Grassmann's day, the only axiomatic theory was Euclidean geometry, and the general notion of an abstract algebra had yet to be defined.

Dehn's proof is an instance in which abstract algebra is used to prove an impossibility result in geometry.

In general, Bourbaki has been criticized for reducing geometry as a whole to abstract algebra and soft analysis.

He regarded geometry as `` the first essential in the training of philosophers ", because of its abstract character.

From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century ; from the initial idea of homology as a topologically invariant relation on chains, the range of applications of homology and cohomology theories has spread out over geometry and abstract algebra.

It is a very broad and abstract generalization of the differential geometry of surfaces in R < sup > 3 </ sup >.

Several major strands of more abstract mathematics ( including invariant theory, the Italian school of algebraic geometry, and Felix Klein's Erlangen programme leading to the study of the classical groups ) built on projective geometry.

Meanwhile, in order to make the proof of the Riemann hypothesis for curves over finite fields that he had announced in 1940 work, he had to introduce the notion of an abstract variety and to rewrite the foundations of algebraic geometry to work with varieties without projective embeddings ( see also the history section in the Algebraic Geometry article ).

* in algebraic geometry, an algebraic variety, which may be affine, projective or abstract

See also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry.