Algebra

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"Algebraist" redirects here. For the novel by Iain M. Banks, see The Algebraist. For beginner's introduction to algebra, see Wikibooks: Algebra. Page semi-protected

The quadratic formula expresses the solution of the degree two equation ax^2 + bx +c=0 in terms of its coefficients a, b, c. Algebra (from Arabic al-jebr meaning "reunion of broken parts"[1]) is one of the broad parts of mathematics, together with number theory, geometry and analysis. 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 origin of its name suggests, was done in the Near East, by such mathematicians as Omar Khayyam (1050-1123). 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.[2] 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. Contents [hide]

1 How to distinguish between different meanings of "algebra" 2 Algebra as a branch of mathematics

3 Etymology

4 History

4.1 Prehistory of algebra

4.2 History of algebra

5 Topics containing the word "algebra"

6 Elementary algebra

6.1 Polynomials

6.2 Teaching algebra

7 Abstract algebra

7.1 Groups

7.2 Rings and fields

8 See also

9 Notes

10 References

11 External links

How to distinguish between different meanings of "algebra"

For historical reasons, the word "algebra" has several related meanings in mathematics, as a single word or with qualifiers. Such a situation, where a single word has many meanings in the same area of mathematics, may be confusing. However the distinction is easier if one recalls that the name of a scientific area is usually singular and without an article and the name of a specific structure requires an article or the plural. Thus we have: As a single word without article, "algebra" names a broad part of mathematics (see below). As a single word with article or in plural, "algebra" denotes a specific mathematical structure. See algebra (ring theory) and algebra over a field. With a qualifier, there is the same distinction:

Without article, it means a part of algebra, such as linear algebra, elementary algebra (the symbol-manipulation rules taught in elementary courses of mathematics as part of primary and secondary education), or abstract algebra (the study of the algebraic structures for themselves). With an article, it means an instance of some abstract structure, like a Lie algebra or an associative algebra. Frequently both meanings exist for the same qualifier, as in the sentence: Commutative algebra is the study of commutative rings, which are commutative algebras over the integers. Algebra as a branch of mathematics

Algebra began with computations similar to those of arithmetic, with letters standing for numbers.[2] This allowed proofs of properties that are true no...