# Areas of Mathematics - Major Divisions of Mathematics - Algebra

Algebra

The study of structure begins with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are recorded in elementary algebra. The deeper properties of these numbers are studied in number theory. The investigation of methods to solve equations leads to the field of abstract algebra, which, among other things, studies rings and fields, structures that generalize the properties possessed by everyday numbers. Long standing questions about compass and straightedge construction were finally settled by Galois theory. The physically important concept of vectors, generalized to vector spaces, is studied in linear algebra.

Order theory
Any set of real numbers can be written out in ascending order. Order Theory extends this idea to sets in general. It includes notions like lattices and ordered algebraic structures. See also the order theory glossary and the list of order topics.
General algebraic systems
Given a set, different ways of combining or relating members of that set can be defined. If these obey certain rules, then a particular algebraic structure is formed. Universal algebra is the more formal study of these structures and systems.
Number theory
Number theory is traditionally concerned with the properties of integers. More recently, it has come to be concerned with wider classes of problems that have arisen naturally from the study of integers. It can be divided into elementary number theory (where the integers are studied without the aid of techniques from other mathematical fields); analytic number theory (where calculus and complex analysis are used as tools); algebraic number theory (which studies the algebraic numbers - the roots of polynomials with integer coefficients); geometric number theory; combinatorial number theory; transcendental number theory; and computational number theory. See also the list of number theory topics.
Field theory and polynomials
Field theory studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined. A polynomial is an expression in which constants and variables are combined using only addition, subtraction, and multiplication.
Commutative rings and algebras
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. This means that if a and b are any elements of the ring, then a×b=b×a. Commutative algebra is the field of study of commutative rings and their ideals, modules and algebras. It is foundational both for algebraic geometry and for algebraic number theory. The most prominent examples of commutative rings are rings of polynomials.

### Other articles related to "algebra, algebras":

Initial Algebra - Use in Computer Science
... used in programming, such as lists and trees, can be obtained as initial algebras of specific endofunctors ... While there may be several initial algebras for a given endofunctor, they are unique up to isomorphism, which informally means that the "observable" properties of a data structure can be adequately ... that the list-forming operations are Combined into one function, they give , which makes this an F-algebra for the endofunctor F sending to ...
Bivector - History
... in 1844 by German mathematician Hermann Grassmann in exterior algebra as the result of the exterior product of two vectors ... Kingdon Clifford in 1888 added the geometric product to Grassmann's algebra, incorporating the ideas of both Hamilton and Grassmann, and founded Clifford algebra, that the bivector as it is known ... Today the bivector is largely studied as a topic in geometric algebra, a Clifford algebra over real or complex vector spaces with a nondegenerate quadratic form ...
List Of Commutative Algebra Topics
... Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings ... Both algebraic geometry and algebraic number theory build on commutative algebra ...
Topological Algebra
... In mathematics, a topological algebra A over a topological field K is a topological vector space together with a continuous multiplication that makes it an algebra over K ... A unital associative topological algebra is a topological ring ... An example of a topological algebra is the algebra C of continuous real-valued functions on the closed unit interval, or more generally any Banach algebra ...
Abstract Algebra - Rings and Fields
... A ring has two binary operations (+) and (×), with × distributive over + ... Under the first operator (+) it forms an abelian group ...