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 mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis.

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

Binary relations are used in many branches of mathematics to model concepts like " is greater than ", " is equal to ", and " divides " in arithmetic, " is congruent to " in geometry, " is adjacent to " in graph theory, " is orthogonal to " in linear algebra and many more.

In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space.

The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics ( for example Venn diagrams and symbolic reasoning about their Boolean algebra ), and the everyday usage of set theory concepts in most contemporary mathematics.

Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, and combinatorics also has many applications in optimization, computer science, ergodic theory and statistical physics.

General category theory, an extension of universal algebra having many new features allowing for semantic flexibility and higher-order logic, came later ; it is now applied throughout mathematics.

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

Mathematics used in rendering includes: linear algebra, calculus, numerical mathematics, signal processing, and Monte Carlo methods.

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 invention of Cartesian coordinates in the 17th century by René Descartes ( Latinized name: Cartesius ) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra.

Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory, and more.

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 Arithmetica is the major work of Diophantus and the most prominent work on algebra in Greek mathematics.

Diophantus ' work created a foundation for work on algebra and in fact much of advanced mathematics is based on algebra.

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.

Elementary algebra introduces the basic rules and operations of algebra, one of the main branches of mathematics.

Elementary algebra is typically taught to secondary school students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic as " algebra ".

This is an unsolved problem which probably has never been seriously investigated, although one frequently hears the comment that we have insufficient specialists of the kind who can compete with the Germans or Swiss, for example, in precision machinery and mathematics, or the Finns in geochemistry.

Next September, after receiving a degree from Yale's Master of Arts in Teaching Program, I will be teaching somewhere -- that much is guaranteed by the present shortage of mathematics teachers.

But because science is based on mathematics doesn't mean that a hot rodder must necessarily be a mathematician.

Like primitive numbers in mathematics, the entire axiological framework is taken to rest upon its operational worth.

In the new situation, philosophy is able to provide the social sciences with the same guidance that mathematics offers the physical sciences, a reservoir of logical relations that can be used in framing hypotheses having explanatory and predictive value.

So, too, is the mathematical competence of a college graduate who has majored in mathematics.

The principal of the school announced that -- despite the help of private tutors in Hollywood and Philadelphia -- Fabian is a 10-o'clock scholar in English and mathematics.

In mathematics and statistics, the arithmetic mean, or simply the mean or average when the context is clear, is the central tendency of a collection of numbers taken as the sum of the numbers divided by the size of the collection.

The term " arithmetic mean " is preferred in mathematics and statistics because it helps distinguish it from other means such as the geometric and harmonic mean.

In addition to mathematics and statistics, the arithmetic mean is used frequently in fields such as economics, sociology, and history, though it is used in almost every academic field to some extent.

The use of the soroban is still taught in Japanese primary schools as part of mathematics, primarily as an aid to faster mental calculation.

In mathematics and computer science, an algorithm ( originating from al-Khwārizmī, the famous Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī ) is a step-by-step procedure for calculations.

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that " the product of a collection of non-empty sets is non-empty ".

The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced.

The axiom of choice has also been thoroughly studied in the context of constructive mathematics, where non-classical logic is employed.

:" A choice function exists in constructive mathematics, because a choice is implied by the very meaning of existence.

Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned.

One of the most interesting aspects of the axiom of choice is the large number of places in mathematics that it shows up.

There is no prize awarded for mathematics, but see Abel Prize.

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative.

In mathematics, specifically commutative algebra, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated.

In mathematics, a unique factorization domain ( UFD ) is a commutative ring in which every non-unit element, with special exceptions, can be uniquely written as a product of prime elements ( or irreducible elements ), analogous to the fundamental theorem of arithmetic for the integers.

In elementary mathematics the coordinates are taken to be real numbers, but may be complex numbers or elements of a more abstract system such as a commutative ring.

In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field ( or, more generally over a commutative ring ) with an extra layer of structure, known as a gradation ( or grading ).

In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams.

In mathematics, and especially in category theory, a commutative diagram is a diagram of objects ( also known as vertices ) and morphisms ( also known as arrows or edges ) such that all directed paths in the diagram with the same start and endpoints lead to the same result by composition.

In mathematics, a ringed space is, intuitively speaking, a space together with a collection of commutative rings, the elements of which are " functions " on each open set of the space.

Algebraic geometry and algebraic number theory, which provide many natural examples of commutative rings, have driven much of the development of commutative ring theory, which is now, under the name of commutative algebra, a major area of modern mathematics.

In commutative ring theory, a branch of mathematics, the radical of an ideal I is an ideal such that an element x is in the radical if some power of x is in I.

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.

Today the commutative property is a well known and basic property used in most branches of mathematics.

In the mathematical fields of topology and K-theory, the Serre – Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules and gives rise to a common intuition throughout mathematics: " projective modules over commutative rings are like vector bundles on compact spaces ".

In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory.

In mathematics, the nine lemma is a statement about commutative diagrams and exact sequences valid in any abelian category, as well as in the category of groups.

In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains.

In mathematics, Serre's multiplicity conjectures, named after Jean-Pierre Serre, are certain purely algebraic problems, in commutative algebra, motivated by the needs of algebraic geometry.

In mathematics, the homological conjectures have been a focus of research activity in commutative algebra since the early 1960s.

In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring ( which may be regarded as a free commutative algebra ).

In mathematics, a Cohen – Macaulay ring is a particular type of commutative ring, possessing some of the algebraic-geometric properties of a nonsingular variety, such as local equidimensionality.

In mathematics, diophantine geometry is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry over a ground field K that is not algebraically closed, such as the field of rational numbers or a finite field, or more general commutative ring such as the integers.