Abstract algebra

Abstract algebra is a branch of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The term "abstract algebra" was coined in the early 20th century to distinguish this area of study from the parts of algebra that deal with the properties of the real number system and the solutions of polynomial equations in one or more variables. Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic to more general concepts.

Overview

Abstract algebra is concerned with the general algebraic structures and the formal properties that these structures possess. It provides a unifying framework for algebraic concepts through the use of axioms and the study of the implications of these axioms. For example, in the study of groups, one of the fundamental structures in abstract algebra, the focus is on sets equipped with a single binary operation that satisfies certain axioms, such as associativity, the existence of an identity element, and the existence of inverse elements.

History

The origins of abstract algebra can be traced back to the work of several mathematicians in the 19th century, including Évariste Galois, who introduced Galois theory to solve polynomial equations and to understand the symmetries of the roots of these equations. The development of abstract algebra was further advanced by mathematicians such as Arthur Cayley, who studied the concept of abstract groups, and Richard Dedekind, who introduced the idea of rings.

Key Concepts

Groups

A group is a set equipped with a binary operation that combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied: closure, associativity, the existence of an identity element, and the existence of inverse elements. Groups are used to represent and study symmetries.

Rings

A ring is a set equipped with two binary operations, usually called addition and multiplication, where addition forms an abelian group, multiplication is associative, and multiplication distributes over addition. Rings generalize the arithmetic properties of the integers.

Fields

A field is a ring in which every non-zero element has a multiplicative inverse. Fields underpin the algebraic structure of the rational, real, and complex number systems, and are central to both algebra and algebraic geometry.

Modules and Vector Spaces

A module is a generalization of the concept of a vector space, wherein the scalars are elements of a given ring instead of a field. A vector space is a collection of vectors, which may be added together and multiplied ("scaled") by numbers, called scalars in this context. Scalars are elements of a field.

Algebras

An algebra over a field is a vector space equipped with a bilinear product. Such structures include familiar examples like complex numbers, quaternions, and matrices.

Applications

Abstract algebra has applications in many areas of mathematics and science. In mathematics, it is used in algebraic geometry, number theory, and topology. Outside of mathematics, it finds applications in physics, particularly in the study of quantum mechanics, and in computer science, especially in cryptography and coding theory.

See Also

• Linear algebra
• Commutative algebra
• Homological algebra
• Universal algebra