Abstract algebra is the subject area of mathematics that studies algebraic structures such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulae and algebraic expressions involving unknowns and real or complex numbers, often now called elementary algebra. The distinction is rarely made in more recent writings.
Contemporary mathematics and mathematical physics make extensive use of abstract algebra; for example, theoretical physics draws on Lie algebras. Subject areas such as algebraic number theory, algebraic topology, and algebraic geometry apply algebraic methods to other areas of mathematics. Representation theory, roughly speaking, takes the 'abstract' out of 'abstract algebra', studying the concrete side of a given structure; see model theory.
Two mathematical subject areas that study the properties of algebraic structures viewed as a whole are universal algebra and category theory. Algebraic structures, together with the associated homomorphisms, form categories. Category theory is a powerful formalism for studying and comparing different algebraic structures.
Read more about Abstract Algebra: History, Basic Concepts, Applications
Famous quotes containing the words abstract and/or algebra:
“Somebody once said that I am incapable of drawing a man, but that I draw abstract things like despair, disillusion, despondency, sorrow, lapse of memory, exile, and that these things are sometimes in a shape that might be called Man or Woman.”
—James Thurber (1894–1961)
“Poetry has become the higher algebra of metaphors.”
—José Ortega Y Gasset (1883–1955)