Branches of Mathematics
Algebra
Also Known As Al-Jabr
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The study of mathematical symbols and the rules for manipulating them: from solving equations for an unknown quantity, the sense the word carried in Al-Khwarizmi's ninth century work that gave algebra its name, to the abstract study of structures such as groups, rings and fields that emerged in the nineteenth century. Modern abstract algebra asks what properties a system of objects and operations must have to behave in a given way, independent of what the objects actually are. Algebraic method itself is older than the name: Babylonian scribes solved quadratic equations numerically more than a thousand years before Al-Khwarizmi, the Greek mathematician Diophantus systematized equation-solving in his third century Arithmetica (some historians credit him, not Al-Khwarizmi, as algebra's true founder, since Al-Khwarizmi's own distinct contribution was a general method rather than the first equation-solving), and the Chinese Nine Chapters on the Mathematical Art, compiled by the first century CE, solved systems of linear equations by a matrix-like elimination method centuries before either. Al-Khwarizmi's systematic, general method independent of specific numbers is why the field bears his book's name rather than an earlier solver's.
Facts
Central QuestionWhat structures of numbers, symbols and operations obey what rules, and what is really required to solve an equation for an unknown? 1 Key DebateWhether abstract algebra's drive toward ever greater generality, groups, rings, fields, categories, trades away the concrete problem-solving intuition (solving this equation, factoring this number) that gave the subject its name and its original motivation. 1 Cross-Tradition Connections
Associated With
Includes
Precise field is algebraic geometry, the study of algebraic cycles on complex varieties; the atlas's taxonomy has no dedicated algebraic-geometry branch, so both its nearest neighbors are recorded.
Sources
1. MacTutor History of Mathematics Archive
University of St Andrews, School of Mathematics and Statisticshttps://mathshistory.st-andrews.ac.uk/HistTopics/Fund_theorem_of_algebra/Quote, https://mathshistory.st-andrews.ac.uk/HistTopics/Fund_theorem_of_algebra/
Every polynomial equation of degree n with complex coefficients has n roots in the complex numbers.
1. MacTutor History of Mathematics Archive
University of St Andrews, School of Mathematics and StatisticsAssociated With: Muhammad ibn Musa al-Khwarizmi, https://mathshistory.st-andrews.ac.uk/Biographies/Al-Khwarizmi/Quote, Associated With: Muhammad ibn Musa al-Khwarizmi, https://mathshistory.st-andrews.ac.uk/Biographies/Al-Khwarizmi/
the first to teach algebra in an elementary form and for its own sake
Wikipedia: Algebra
Wikimedia FoundationLead section, opening paragraphQuote, Lead section, opening paragraph
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems.
Lagrange's Theorem, Group Theory (Wikipedia)
Wikimedia FoundationIncludes: Lagrange's Theorem (Group Theory), opening sentenceQuote, Includes: Lagrange's Theorem (Group Theory), opening sentence
Lagrange's theorem states that if H is a subgroup of any finite group G, then |H| is a divisor of |G|.
Vector Space (Wikipedia)
Wikimedia FoundationIncludes: Vector Space, lead paragraphQuote, Includes: Vector Space, lead paragraph
A vector space (also called a linear space) is a set whose elements, often called vectors, can be added together and multiplied ("scaled") by numbers called scalars.
Ring, Mathematics (Wikipedia)
Wikimedia FoundationIncludes: Ring (Abstract Algebra), lead paragraphQuote, Includes: Ring (Abstract Algebra), lead paragraph
In mathematics, a ring is an algebraic structure consisting of a set with two binary operations typically called addition and multiplication
Stone-Weierstrass Theorem (Wikipedia)
WikipediaIncludes: Field (Abstract Algebra), lead paragraphQuote, Includes: Field (Abstract Algebra), lead paragraph
A field is an algebraic structure that is closed under the four usual arithmetic operations.
Complex Number (Wikipedia)
Wikimedia FoundationIncludes: Complex Number, lead paragraphQuote, Includes: Complex Number, lead paragraph
Every complex number can be expressed in the form a + bi, where a and b are real numbers
Algebraic Geometry (Wikipedia)
Wikimedia FoundationAssociated With: Algebraic Geometry, lead paragraphQuote, Associated With: Algebraic Geometry, lead paragraph
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems.
Polynomial (Wikipedia)
Wikimedia FoundationIncludes: Polynomial, Introductory sectionQuote, Includes: Polynomial, Introductory section
In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry.
Equation (Wikipedia)
Wikimedia FoundationIncludes: Equation, Lead sectionQuote, Includes: Equation, Lead section
Algebra studies two main families of equations: polynomial equations and, among them, the special case of linear equations.
Wikipedia: Cayley's Theorem
Fundamental Theorem of Algebra (Wikipedia)
Wikimedia FoundationIncludes: Fundamental Theorem of Algebra, History sectionQuote, Includes: Fundamental Theorem of Algebra, History section
it was named when algebra was synonymous with the theory of equations
Hodge Conjecture (Wikipedia)
Wikimedia FoundationIncludes: Hodge Conjecture, Opening paragraphQuote, Includes: Hodge Conjecture, Opening paragraph
a major unsolved problem in algebraic geometry and complex geometry
Wikipedia: Constructible Polygon
Wikimedia FoundationIncludes: Constructibility of the Regular Heptadecagon, Constructibility section, algebraic criterion behind Gauss's proofQuote, Includes: Constructibility of the Regular Heptadecagon, Constructibility section, algebraic criterion behind Gauss's proof
Gauss's proof relies firstly on the fact that constructibility is equivalent to expressibility of the trigonometric functions of the common angle in terms of arithmetic operations and square root extractions
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