29 articles
Abstract algebra
Abstract algebra, also called modern algebra, is the branch of mathematics that studies algebraic structures like groups, rings, and fields, rather than computing with particular number systems.
Al-Khwarizmi (محمد بن موسى الخوارزمي)
Al-Khwarizmi (Muhammad ibn Musa al-Khwarizmi, ca. 780–850) was a mathematician and astronomer at the House of Wisdom in Baghdad whose algebra treatise gave us "algebra", and whose Latinized name gave "algorithm".
Algebra
Algebra, historically called the cossic art, is a branch of mathematics studying operations and structures, from solving equations with unknowns to abstract algebra's groups, rings, and fields.
Binary operation
A binary operation, also called a binary operator or dyadic operation, is a function combining two elements of a set to yield an element of the same set.
Binomial theorem
The binomial theorem, or binomial formula, expands powers of a sum using binomial coefficients from Pascal's triangle; first attributed to al-Karaji around 1029 and extended to fractional powers by Newton.
Degree of a polynomial
The degree of a polynomial, or polynomial degree, is the largest exponent with a nonzero coefficient among its terms, naming it linear, quadratic, cubic, and so on.
Direct product
The direct product is a construction in algebra that combines structures of the same kind, such as groups or rings, into a single structure with coordinatewise operations.
Discriminant
The discriminant is a quantity computed from the coefficients of a polynomial or quadratic form that vanishes exactly when the object degenerates, as when two roots coincide.
Elementary algebra
Elementary algebra, also known as college algebra or high school algebra, is the branch of mathematics that uses letters as variables to write and solve equations, taught in school and introductory college courses.
Emmy Noether
Emmy Noether, born Amalie Emmy Noether, was a German mathematician whose 1918 theorems linked symmetries to conservation laws, and whose 1921 ideal theory founded modern abstract algebra.
Equation
An equation is a mathematical statement that two expressions are equal, written with an equals sign; identities hold for all values, conditional equations for some, and contradictions for none.
Equivalence relation
An equivalence relation is a binary relation that is reflexive, symmetric, and transitive, so it behaves like equality; its equivalence classes form a partition of the set.
Gerolamo Cardano
Gerolamo Cardano, also Girolamo Cardano, was an Italian physician, mathematician, and astrologer whose 1545 Ars Magna gave the first published solutions of cubic and quartic equations.
Green's relations
Green's relations are five equivalence relations defined on a semigroup from its principal ideals, introduced by J. A. Green in 1951 and central to semigroup theory.
History of algebra
The history of algebra traces techniques for finding unknown quantities, from Babylonian tablets around 1800 BC through al-Khwarizmi, who gave the subject its name, to the abstract structures of the twentieth century.
Homomorphism
A homomorphism is a map between two algebraic structures of the same kind, such as groups or rings, that preserves their operations, making it algebra's basic tool for comparing structures.
Inverse semigroup
An inverse semigroup is a semigroup in which every element has a unique inverse, the algebraic structure capturing partial symmetries, defined independently by Vagner and Preston in the 1950s.
Linear equation
A linear equation, also called a first-degree equation, is an algebraic equation where each variable appears only to the first power, as in ax + b = 0.
List of publications in mathematics
The list of publications in mathematics is a curated record of the mathematical literature, from Euclid's Elements of 300 BC to modern reviewing databases like MathSciNet and zbMATH Open.
Nicolo Tartaglia
Nicolo Tartaglia, born Niccolo Fontana, was an Italian mathematician and engineer who solved a class of cubic equations and wrote the first printed treatise on ballistics.
Polynomial
A polynomial is a mathematical expression built from powers of one or more variables multiplied by coefficients, forming the basic object of algebra and a key tool in cryptography and numerical analysis.
Quadratic equation
A quadratic equation, also called a second degree equation, is a polynomial equation of degree two, written ax² + bx + c = 0 and solved by the quadratic formula.
Quartic function
A quartic function is a polynomial function of degree four, solved generally by Lodovico Ferrari's method in 1540. It is the highest degree solvable by radicals.
Rational function
A rational function, also called a fractional-rational function, is a function expressible as the ratio of two polynomials; such ratios form the field of rational functions F(x) in abstract algebra.
Regular semigroup
A regular semigroup is a semigroup in which every element a has an inverse b satisfying a = aba, including groups, inverse semigroups, and matrix semigroups.
Semigroup
A semigroup is an algebraic structure consisting of a non-empty set with an associative binary operation, sitting between a magma and a monoid in the hierarchy of algebraic structures.
Transformation semigroup
A transformation semigroup is a set of functions from a set to itself that is closed under composition; with the identity function it is a transformation monoid.
Unary operation
A unary operation, also called a unary or monadic operator, is a function with exactly one input, such as negation, square root, or multiplicative inverse in a group.
Vieta's formulas
Vieta's formulas relate a polynomial's coefficients to the sums and products of its roots; for ax² + bx + c, the roots add to −b/a and multiply to c/a.