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Commutative Algebra and Algebraic Geometry

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The algebra , the Arabic al-jabr (الجبر) meaning "reduction of a fracture", "food", is a branch of mathematics . This name comes from a book by the $IX^{th}$ century , Kitab al-Jabr wa'l-muqabalah ("Book of the restoration and comparison"), due to the mathematician Al-Khwarizmi . In a first approach, the algebra can be defined as a discipline systematizing methods of solving mathematical problems. Its scope extends arithmetic problems dealing with numbers, those of geometric origin. And algebra occupies a pivotal position between arithmetic and geometry lead to extend the digital domain. Depending on the time or level of education considered, algebra can be described as: an arithmetic widespread, extending to different objects or variables the operations of the usual numbers, the theory of equations and polynomials , in more advanced and from the $XX^{th}$ century , the study of certain structures algebraic. Often associated with the calculation of the letters $x$ , $y$ , it also provides the basis of mathematical language commonly used in different sciences. It also assigns the epithet of "algebraic" to other parts of mathematics, including objects or methods within the algebra and the algebraic geometry is the part of the geometry that studies curves or algebraic varieties , that is to say, curves or varieties defined by polynomial equations with techniques themselves often from algebra.
Mohamed AQALMOUN Professeur de l'enseignement supérieur ENS-Fès