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 rare cases a linguist may represent phonemes with abstract symbols, such as dingbats, so as not to privilege any one allophone.

The logo was once again changed in 1990 into an abstract version of the previous one.

In computer science, an abstract data type ( ADT ) is a mathematical model for a certain class of data structures that have similar behavior ; or for certain data types of one or more programming languages that have similar semantics.

Then one of the prelates of the upper bar made an abstract, and another prelate of the same bar revised it.

By contrast, in mainstream Analytical philosophy the topic is more confined to abstract investigation, in the work of such influential theorists as W. V. O. Quine, to name one of many.

Because these two definitions can be transformed simply by into the other, some formulae have this alternatingly (- 1 )< sup > n </ sup >- term and others not depending on the context, but it is not possible to decide in favor of one of these definitions to be the correct or appropriate or natural one ( for the abstract Bernoulli numbers ).

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.

Paradoxically, achieving this connective human quality has also moved his buildings away from the abstract imageability valued in contemporary architecture, and this is one reason why his buildings are under-appreciated at present.

Although many scholars today consider the Treatise to be Hume's most important work and one of the most important books in Western philosophy, the critics in Great Britain at the time did not agree, describing it as " abstract and unintelligible ".

This differs from the queue abstract data type or First-In-First-Out List ( FIFO ), where elements can only be added to one end and removed from the other.

For Avicenna ( Ibn Sina ), for example, the a tabula rasa is a pure potentiality that is actualized through education, and knowledge is attained through " empirical familiarity with objects in this world from which one abstracts universal concepts " developed through a " syllogistic method of reasoning in which observations lead to propositional statements which when compounded lead to further abstract concepts.

" Such definitions depend upon "( cultural ) processes rather than abstract musical types ...", upon " continuity and oral transmission ... seen as characterizing one side of a cultural dichotomy, the other side of which is found not only in the lower layers of feudal, capitalist and some oriental societies but also in ' primitive ' societies and in parts of ' popular cultures '.

It is conceived as an abstract machine that can be in one of a finite number of states.

On Pappas reading, Berkeley ’ s two theses — that there are no abstract ideas and that sensible objects must be perceived in order to exist — entail one another.

*" Fine-grained ", designating a more finely differentiated form of a technical or abstract system, in contrast to " coarse-grained ", a less differentiated one

The abstraction provided by group actions is a powerful one, because it allows geometrical ideas to be applied to more abstract objects.

The heap is one maximally efficient implementation of an abstract data type called a priority queue.

He chose Louis Van Lint, one of the most respected Belgian abstract painters at the time, whose work he liked a lot, to be his private teacher.

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.

By lumping the tenants into one abstract entity, the argument renders itself vulnerable to a slippery slope argument.

However, because these tools do not implement the " web of abstract concepts " hiding behind the system of natural-language macros, or provide an ability to change the order of the source code from a machine-imposed sequence to one convenient to the human mind, they cannot properly be called literate programming tools in the sense intended by Knuth.

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