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The word "algebra" is derived from the Arabic word الجبر al-jabr, and this comes from the treatise written in the year 830 by the medieval Persian mathematician, Al-Khwārizmī, whose Arabic title, Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala, can be translated as The Compendious Book on Calculation by Completion and Balancing.
Algebra is the branch of mathematics that studies certain abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication.
The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra, as established by the influence and works of Emmy Noether. [36] Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics.
Al-Khwarizmi's algebra is regarded as the foundation and cornerstone of the sciences. In a sense, al-Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because al-Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers. [52]
Algebraic functions are functions that can be expressed as the solution of a polynomial equation with integer coefficients.. Polynomials: Can be generated solely by addition, multiplication, and raising to the power of a positive integer.
Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers. It is typically taught to secondary school students and at introductory college level in the United States, [5] and builds on their understanding of arithmetic. The use of variables to denote quantities allows ...
Hilbert's basis theorem (commutative algebra,invariant theory) Hilbert's Nullstellensatz (theorem of zeroes) (commutative algebra, algebraic geometry) Hilbert–Schmidt theorem (functional analysis) Hilbert–Speiser theorem (cyclotomic fields) Hilbert–Waring theorem (number theory) Hilbert's irreducibility theorem (number theory)
Noether grew up in the Bavarian city of Erlangen, depicted here in a 1916 postcard.. Amalie Emmy Noether was born on 23 March 1882 in Erlangen, Bavaria. [13] She was the first of four children of mathematician Max Noether and Ida Amalia Kaufmann, both from wealthy Jewish merchant families. [14]
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