## Algebra is a branch of mathematics connected with the study of structure, relation, and quantity.

The origin of Bij-ganitam is ancient India. Persian first learn it from here and then from them the Greeks. The Algebra name is derived from the treatise written by the Persian mathematician Muhammad bin Mūsā al-Khwārizmī 820CE, titled (in Arabic كتاب الجبر والمقابلة )Al-Kitab al-Jabr wa-l-Muqabala (meaning “The Compendious Book on Calculation by Completion and Balancing”)

## Major fields of mathematics :

Logic • Set theory • Algebra (Abstract algebra – Linear algebra) • Discrete mathematics • Combinatorics • Number theory • Analysis • Geometry • Topology • Applied mathematics • Probability • Statistics • Mathematical physics • Ring (mathematics) and Field (mathematics)

**Elementary and Abstract Algebra**

Elementary algebra is the most basic form of algebra. It is taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. Although in arithmetic, only numbers and their arithmetical operations (such as +, −, ×, ÷) occur, in algebra, numbers are often denoted by symbols (such as a, x, y).

Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic of numbers to more general concepts.

**The word algebra is also used for various algebraic structures:**

Algebra over a field

Algebra over a set

Boolean algebra

F-algebra and F-coalgebra in category theory

Sigma-algebra

## Fields of algebra

» Abstract algebra » Affine Hecke algebra » Algebra

» Algebra homomorphism » Algebra stubs » Algebraic element

» Algebraic extension » Algebraic function » Algebraic geometry

» Algebraic number » Algebraic solution » Algebraic topology

» Alternatization » Anyonic Lie algebra » Associative algebra

» Automorphism » Binary operation » Binomial

» Binomial theorem » Brahmagupta-Fibonacci identity » Canonical form

» Category theory » Closed-form expression » Coefficient

» Combinatorics » Completing the square » Computer algebra

» Congruence relation » Conjugate (algebra) » Consequence operator

» Cuntz algebra » Cycle (mathematics) » Degree (mathematics)

» Derivative (generalizations) » Determinant » Digital root

» Dimension » Distributive homomorphism » Distributive lattice

» Elementary algebra » Expression (mathematics) » Factorization

» Filtration (mathematics) » Galois theory » Generalized arithmetic progression

» Goursat’s lemma » Hall polynomial » Hecke algebra

» Homological algebra » Identity element » Immanant of a matrix

» Indeterminate (variable) » Intersection (set theory) » Inverse element

» Irreducible polynomial » Isomorphism » Iterated binary operation

» K-theory » K-theory (physics) » Kernel (algebra)

» Kernel (set theory) » Lattice (order) » Laws of Form

» List of basic algebra topics » Mathematical identities » Monomial

» Monomial basis » Multinomial theorem » Multiplicative inverse

» Nested radical » Operand » Operation theory

» Operator » Order of operations » Pairing

» Partial fraction » Partial fraction decomposition » Perfect square

» Permanent » Permutations » Plugging in (algebra)

» Polynomials » Power set » Ratio

» Rational root theorem » Recurrence relation » Relation algebra

» Ring theory » Square (algebra) » Sylvester’s determinant theorem

» Symbolic method » Symmetric difference » Symmetric functions

» System of linear equations » Temperley-Lieb algebra » Theory of equations

» Topological module » Transforming polynomials » Trinomial

» Union (set theory) » Unital » Universal algebra

» Variable » Variety (universal algebra)

** More on Algebra**

Derivative algebra (abstract algebra) Map algebra F-algebra

The Algebra of Ice C*-algebra *-algebra
MV-algebra Algebra Pre-algebra
Max-plus algebra Rng (algebra) B*-algebra

En (Lie algebra) Lie algebra Abstract algebra

Steenrod algebra Derivative algebra Hopf algebra

Multilinear algebra Group algebra Elements of Algebra

Hecke algebra Modular Lie algebra Graded algebra

De Morgan algebra Heyting algebra Plugging in (algebra)

Graded Lie algebra Elementary algebra Jordan algebra

Monster Lie algebra Linear algebra Operator algebra

Relation algebra Poisson algebra Free Lie algebra

Example of a non-associative algebra Characteristic (algebra) Magma (algebra)

Quaternion algebra Exterior algebra Cuntz algebra

Spectrum of a C*-algebra Strip algebra Topological algebra

Cube (algebra) Matrix algebra Enveloping algebra

Chronology of key algebraic developments :

Circa 2800 BC: Bodhayan Collected old Theories and compiled in Sulva Sutram [India]. It explained so-called Pythagorean triples algebraically, finds geometric solutions of linear equations and quadratic equations of the forms ax2 = c and ax2 + bx = c, and finds two sets of positive integral solutions to a set of simultaneous Diophantine equations.

Circa 1800 BC: The Old Babylonian Strassburg tablet seeks the solution of a quadratic elliptic equation.

Circa 1600 BC: The Plimpton 322 tablet gives a table of Pythagorean triples in Babylonian Cuneiform script.

Circa 600 BC: Indian mathematician Apastamba, in his Apastamba Sulba Sutra, solves the general linear equation and uses simultaneous Diophantine equations with up to five unknowns.

Circa 300 BC: In Book II of his Elements, Euclid gives a geometric construction with Euclidean tools for the solution of the quadratic equation for positive real roots. The construction is due to the Pythagorean School of geometry.

Circa 300 BC: A geometric construction for the solution of the cubic is sought (doubling the cube problem). It is now well known that the general cubic has no such solution using Euclidean tools.

Circa 100 BC: Algebraic equations are treated in the Chinese mathematics book Jiuzhang suanshu (The Nine Chapters on the Mathematical Art), which contains solutions of linear equations solved using the rule of double false position, geometric solutions of quadratic equations, and the solutions of matrices equivalent to the modern method, to solve systems of simultaneous linear equations.

Circa 100 BC: The Bakhshali Manuscript written in ancient India uses a form of algebraic notation using letters of the alphabet and other signs, and contains cubic and quartic equations, algebraic solutions of linear equations with up to five unknowns, the general algebraic formula for the quadratic equation, and solutions of indeterminate quadratic equations and simultaneous equations.

Circa 150 AD: Hellenized Egyptian mathematician Hero of Alexandria, treats algebraic equations in three volumes of mathematics.

Circa 200: Hellenized Babylonian mathematician Diophantus, who lived in Egypt and is often considered the “father of algebra”, writes his famous Arithmetica, a work featuring solutions of algebraic equations and on the theory of numbers.

499: Indian mathematician Aryabhata, in his treatise Aryabhatiya, obtains whole-number solutions to linear equations by a method equivalent to the modern one, describes the general integral solution of the indeterminate linear equation and gives integral solutions of simultaneous indeterminate linear equations.

Circa 625: Chinese mathematician Wang Xiaotong finds numerical solutions of cubic equations.

628: Indian mathematician Brahmagupta, in his treatise Brahma Sputa Siddhanta, invents the chakravala method of solving indeterminate quadratic equations, including Pell’s equation, and gives rules for solving linear and quadratic equations.

820: The word algebra is derived from operations described in the treatise written by the Persian mathematician Muḥammad ibn Mūsā al-Ḵwārizmī titled Al-Kitab al-Jabr wa-l-Muqabala (meaning “The Compendious Book on Calculation by Completion and Balancing”) on the systematic solution of linear and quadratic equations. Al-Khwarizmi is often considered as the “father of algebra”, much of whose works on reduction was included in the book and added to many methods we have in algebra now.

Circa 850: Persian mathematician al-Mahani conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra.

Circa 850: Indian mathematician Mahavira solves various quadratic, cubic, quartic, quintic and higher-order equations, as well as indeterminate quadratic, cubic and higher-order equations.

Circa 990: Persian Abu Bakr al-Karaji, in his treatise al-Fakhri, further develops algebra by extending Al-Khwarizmi’s methodology to incorporate integral powers and integral roots of unknown quantities. He replaces geometrical operations of algebra with modern arithmetical operations, and defines the monomials x, x2, x3, … and 1/x, 1/x2, 1/x3, … and gives rules for the products of any two of these.

Circa 1050: Chinese mathematician Jia Xian finds numerical solutions of polynomial equations.

1072: Persian mathematician Omar Khayyam develops algebraic geometry and, in the Treatise on Demonstration of Problems of Algebra, gives a complete classification of cubic equations with general geometric solutions found by means of intersecting conic sections.

1114: Indian mathematician Bhaskara, in his Bijaganita (Algebra), recognizes that a positive number has both a positive and negative square root, and solves various cubic, quartic and higher-order polynomial equations, as well as the general quadratic indeterminant equation.

1202: Algebra is introduced to Europe largely through the work of Leonardo Fibonacci of Pisa in his work Liber Abaci.

Circa 1300: Chinese mathematician Zhu Shijie deals with polynomial algebra, solves quadratic equations, simultaneous equations and equations with up to four unknowns, and numerically solves some quartic, quintic and higher-order polynomial equations.

Circa 1400: Indian mathematician Madhava of Sangamagramma finds iterative methods for approximate solution of non-linear equations.

1535: Nicolo Fontana Tartaglia and others mathematicians in Italy independently solved the general cubic equation.

1545: Girolamo Cardano publishes Ars magna -The great art which gives Fontana’s solution to the general quartic equation.

1572: Rafael Bombelli recognizes the complex roots of the cubic and improves current notation.

1591: Francois Viete develops improved symbolic notation for various powers of an unknown and uses vowels for unknowns and consonants for constants in In artem analyticam isagoge.

1631: Thomas Harriot in a posthumus publication uses exponential notation and is the first to use symbols to indicate “less than” and “greater than”.

1682: Gottfried Wilhelm Leibniz develops his notion of symbolic manipulation with formal rules which he calls characteristica generalis.

1680s: Japanese mathematician Kowa Seki, in his Method of solving the dissimulated problems, discovers the determinant, and Bernoulli numbers.

1750: Gabriel Cramer, in his treatise Introduction to the analysis of algebraic curves, states Cramer’s rule and studies algebraic curves, matrices and determinants.

1824: Niels Henrik Abel proved that the general quintic equation is insoluble by radicals.

1832: Galois theory is developed by Évariste Galois in his work on abstract algebra.

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