However, European mathematicians, for the most part, resisted these concepts until the middle of the 19th century. The formal study of solving symbolic equations led Leonhard Euler to accept what were then considered "nonsense" roots such as negative numbers and imaginary numbers, in the late 18th century. Fully symbolic algebra did not appear until François Viète's 1591 New Algebra, and even this had some spelled out words that were given symbols in Descartes's 1637 La Géométrie. Muhammad ibn Mūsā al-Khwārizmī originated the word "algebra" in 830 AD, but his work was entirely rhetorical algebra. This word problem stage is classified as rhetorical algebra and was the dominant approach up to the 16th century. Circa 1700 BC, the Babylonians were able to solve quadratic equations specified as word problems. The study of polynomial equations or algebraic equations has a long history. This historical development is almost the opposite of the treatment found in popular textbooks, such as van der Waerden's Moderne Algebra, which start each chapter with a formal definition of a structure and then follow it with concrete examples. This unification occurred in the early decades of the 20th century and resulted in the formal axiomatic definitions of various algebraic structures such as groups, rings, and fields. Most theories that are now recognized as parts of abstract algebra started as collections of disparate facts from various branches of mathematics, acquired a common theme that served as a core around which various results were grouped, and finally became unified on a basis of a common set of concepts. Concrete problems and examples came from number theory, geometry, analysis, and the solutions of algebraic equations. Abstract algebra came into existence during the nineteenth century as more complex problems and solution methods developed. For example, the structure of groups is a single object in universal algebra, which is called the variety of groups.īefore the nineteenth century, algebra meant the study of the solution of polynomial equations.
Universal algebra is a related subject that studies types of algebraic structures as single objects.
Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. The term abstract algebra was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning.Īlgebraic structures, with their associated homomorphisms, form mathematical categories. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. The permutations of the Rubik's Cube form a group, a fundamental concept within abstract algebra.