Portal:Algebra
Introduction
Algebra (from Arabic "al-jabr", literally 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 mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost 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 generally considered to be 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.
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 the letter is unknown, but the law of inverses can be used to discover its value: . In E = mc^{2}, the letters and are variables, and the letter is a constant, the speed of light in a vacuum. Algebra gives methods for writing formulas and solving equations that are much clearer and easier 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.
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The circle of center 0 and radius 1 in the complex plane is a Lie group with complex multiplication. |
In mathematics, a Lie group (/ˈliː/, sounds like "Lee"), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. They are named after the nineteenth century Norwegian mathematician Sophus Lie, who laid the foundations of the theory of continuous transformation groups.
Lie groups represent the best developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. They provide a natural framework for analysing the continuous symmetries of differential equations (Differential Galois theory), much in the same way as permutation groups are used in Galois theory for analysing the discrete symmetries of algebraic equations. An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations, his idée fixe.
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Did you know?
- ...that it is impossible to devise a single formula involving only polynomials and radicals for solving an arbitrary quintic equation?
- ...that it is possible for a three-dimensional figure to have a finite volume but infinite surface area? An example of this is Gabriel's Horn.
- ...that the Gudermannian function relates the regular trigonometric functions and the hyperbolic trigonometric functions without the use of complex numbers?
- ...that the classification of finite simple groups was not completed until the mid 1980s?
- ...that a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers?
- ...that the classification of finite simple groups is of more than 10000 pages.
Topics in algebra
General | Elementary algebra | Key concepts | Linear algebra |
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Algebraic structures | Groups | Rings and Fields | Other |
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Algebra | Analysis |
Category theory |
Computer science |
Cryptography |
Discrete mathematics |
Logic | Mathematics |
Number theory |
Physics | Science | Set theory | Statistics |