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.

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

In mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain ( also called a Euclidean ring ) is a ring that can be endowed with a certain structure – namely a Euclidean function, to be described in detail below – which allows a suitable generalization of the Euclidean division of the integers.

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, an integral domain is a commutative ring that has no zero divisors, and which is not the trivial ring

In abstract algebra, a monoid ring is a new ring constructed from some other ring and a monoid.

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.

In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication.

In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec ( R ), is the set of all proper prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space.

In abstract algebra, a congruence relation ( or simply congruence ) is an equivalence relation on an algebraic structure ( such as a group, ring, or vector space ) that is compatible with the structure.

From the point of view of abstract algebra, congruence modulo is a congruence relation on the ring of integers, and arithmetic modulo occurs on the corresponding quotient ring.

In the branch of mathematics known as abstract algebra, a ring is an algebraic concept abstracting and generalizing the algebraic structure of the integers, specifically the two operations of addition and multiplication.

In abstract algebra, two nonzero elements and of a ring are respectively called a left zero divisor and a right zero divisor if ; this is a partial case of divisibility in rings.

It occurs in the proofs of several theorems of crucial importance, for instance the Hahn – Banach theorem in functional analysis, the theorem that every vector space has a basis, Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems in abstract algebra that every nonzero ring has a maximal ideal and that every field has an algebraic closure.

Van Emde Boas observes " even if we base complexity theory on abstract instead of concrete machines, arbitrariness of the choice of a model remains.

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

Category theory deals with abstract objects and morphisms between those objects.

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

On a more abstract level, model theoretic arguments hold that a given set of symbols in a theory can be mapped onto any number of sets of real-world objects — each set being a " model " of the theory — providing the interrelationships between the objects are the same.

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

Some, such as computational complexity theory, which studies fundamental properties of computational problems, are highly abstract, while others, such as computer graphics, emphasize real-world applications.

Some fields, such as computational complexity theory ( which explores the fundamental properties of computational problems ), are highly abstract, whilst fields such as computer graphics emphasise real-world applications.

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows ( also called morphisms, although this term also has a specific, non category-theoretical meaning ), where these collections satisfy some basic conditions.

Topos theory is a form of abstract sheaf theory, with geometric origins, and leads to ideas such as pointless topology.

Category theory is also, in some sense, a continuation of the work of Emmy Noether ( one of Mac Lane's teachers ) in formalizing abstract processes ; Noether realized that in order to understand a type of mathematical structure, one needs to understand the processes preserving that structure.

Modern European conservatives such as Edmund Burke have found the extreme idealism of either democracy may endanger broader liberties, and similarly reject " abstract reason " as a guide for political theory.

In computational complexity theory, a problem refers to the abstract question to be solved.

Dialectics as represented in dialectical materialism is a theory of the general and abstract factors that govern the development of nature, society, and thought.

Directed sets also give rise to direct limits in abstract algebra and ( more generally ) category theory.

His work laid the foundations for Galois theory and group theory, two major branches of abstract algebra, and the subfield of Galois connections.