Algebra (from Arabic al-jebr meaning “reunion of broken parts”) is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form algebra is the study of symbols and the rules for manipulating symbols and is a unifying thread of all of mathematics. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields.

The more basic parts of algebra are called elementary algebra, the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Much early work in algebra, as the Arabic origin of its name suggests, was done in the Near East, by such mathematicians as Omar Khayyam (1048-1131).

Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values. For example, in x + 2 = 5 the letter x is unknown, but the law of inverses can be used to discover its value: x=3. In E=mc^2, the letters E and m are variables, and the letter c is a constant. Algebra gives methods for solving equations and expressing formulas that are much easier (for those who know how to use them) than the older method of writing everything out in words.

The word algebra is also used in certain specialized ways. A special kind of mathematical object in abstract algebra is called an “algebra”, and the word is used, for example, in the phrases linear algebra and algebraic topology (see below).

A mathematician who does research in algebra is called an algebraist.

Algebra as a branch of mathematics
Algebra began with computations similar to those of arithmetic, with letters standing for numbers.[6] This allowed proofs of properties that are true no matter which numbers are involved. For example, in the quadratic equation

a, b, c can be any numbers whatsoever (except that a cannot be 0), and the quadratic formula can be used to quickly and easily find the value of the unknown quantity x.

As it developed, algebra was extended to other non-numerical objects, such as vectors, matrices, and polynomials. Then the structural properties of these non-numerical objects were abstracted to define algebraic structures such as groups, rings, and fields.

Before the 16th century, mathematics was divided into only two subfields, arithmetic and geometry. Even though some methods, which had been developed much earlier, may be considered nowadays as algebra, the emergence of algebra and, soon thereafter, of infinitesimal calculus as subfields of mathematics only dates from 16th or 17th century. From the second half of 19th century on, many new fields of mathematics appeared, most of which made use of both arithmetic and geometry, and almost all of which used algebra.

Today, algebra has grown until it includes many branches of mathematics, as can be seen in the Mathematics Subject Classification where none of the first level areas (two digit entries) is called algebra. Today algebra includes section 08-General algebraic systems, 12-Field theory and polynomials, 13-Commutative algebra, 15-Linear and multilinear algebra; matrix theory, 16-Associative rings and algebras, 17-Nonassociative rings and algebras, 18-Category theory; homological algebra, 19-K-theory and 20-Group theory. Algebra is also used extensively in 11-Number theory and 14-Algebraic geometry.

The word algebra comes from the Arabic language (الجبر al-jabr “restoration”) from the title of the book Ilm al-jabr wa’l-muḳābala by al-Khwarizmi. The word entered the English language during Late Middle English from either Spanish, Italian, or Medieval Latin. Algebra originally referred to a surgical procedure, and still is used in that sense in Spanish, while the mathematical meaning was a later development.

The start of algebra as an area of mathematics may be dated to the end of 16th century, with François Viète’s work. Until the 19th century, algebra consisted essentially of the theory of equations. In the following, “Prehistory of algebra” is about the results of the theory of equations that precede the emergence of algebra as an area of mathematics.