Portal:Algebra

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Algebra is a branch of mathematics concerning the study of structure, relation and quantity. The name is derived from the treatise written by the Persian mathematician, astronomer, astrologer and geographer, Muḥammad ibn Mūsā al-Khwārizmī titled Kitab al-Jabr al-Muqabala (meaning "The Compendious Book on Calculation by Completion and Balancing"), which provided operations for the systematic solution of linear and quadratic equations.

Together with geometry, analysis, combinatorics, and number theory, algebra is one of the main branches of mathematics. Elementary algebra is often part of the curriculum in secondary education and provides an introduction to the basic ideas of algebra, including effects of adding and multiplying numbers, the concept of variables, definition of polynomials, along with factorization and determining their roots.

In addition to working directly with numbers, algebra covers working with symbols, variables, and set elements. Addition and multiplication are viewed as general operations, and their precise definitions lead to structures such as groups, rings and fields. Template:/box-footer

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A loxodromic representation of the Riemann sphere.

The Riemann sphere is a way of extending the plane of complex numbers with one additional point at infinity, in a way that makes expressions such as

\frac{1}{0} = \infty

well-behaved and useful, at least in certain contexts. It is named after 19th century mathematician Bernhard Riemann. It is also called the complex projective line, denoted CP1.

On a purely algebraic level, the complex numbers with an extra infinity element constitute a number system known as the extended complex numbers. Arithmetic with infinity does not obey all of the usual rules of algebra, and so the extended complex numbers do not form a field. However, the Riemann sphere is geometrically and analytically well-behaved, even near infinity; it is a one-dimensional complex manifold, also called a Riemann surface.

In complex analysis, the Riemann sphere facilitates an elegant theory of meromorphic functions. The Riemann sphere is ubiquitous in projective geometry and algebraic geometry as a fundamental example of a complex manifold, projective space, and algebraic variety. It also finds utility in other disciplines that depend on analysis and geometry, such as quantum mechanics and other branches of physics.

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These are all the connected Dynkin diagrams, which classify the irreducible root systems, which themselves classify simple complex Lie algebras and simple complex Lie groups. These diagrams are therefore fundamental throughout Lie group theory.

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