Field (mathematics)

Definition

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational numbers do. A field is thus a fundamental algebraic structure that is widely used in algebra, number theory, and many other areas of mathematics.

Related concepts

Abelian extension Abelian group Abel–Ruffini theorem Absolute Galois group Absolute value Abstract algebraic variety Academic Press Addison-Wesley Addition Additive group Additive identity Additive inverse Adjunction (field theory)Affine space Alexandre-Théophile Vandermonde Algebra Algebra over a field Algebraic K-theory Algebraic closure Algebraic curve Algebraic element Algebraic extension Algebraic function field Algebraic geometry Algebraic number Algebraic number field Algebraic number theory Algebraic structure Algebraic topology Algebraically closed Algebraically independent Allyn and Bacon Alternative algebra Angle trisection Archimedean field Arithmetic geometry Arity Artin–Schreier theorem Associative algebra Associativity Axiom Axiom of choice Ax–Kochen theorem Basis (linear algebra)Bialgebra Bijection Binary field Binary operation Binomial coefficient Binomial formula Birational equivalence Birational geometry Boolean algebra (structure)Bounded set Brauer group Bulletin of the American Mathematical Society Calculus Cambridge University Press Cardinality Carl Friedrich Gauss Carry (arithmetic)Cartesian coordinates Central simple algebra Characteristic (algebra)Charles Hermite Class field theory Coding theory Coefficient Combinatorics Commutative ring Commutativity Compact topological space Compass (drawing tool)Compass and straightedge Complemented lattice Completion (metric space)Complex manifold Complex multiplication Complex number Complex p-adic number Composite number Composition algebra Compositum Computing Constructible numbers Constructive mathematics Continuous map Cryptography Cubic polynomial Cyclic group Cyclotomic field David Eisenbud Dedekind-complete Dedekind cut Degree of a field extension Degree of a polynomial Derivation (abstract algebra)Determinant (mathematics)Differential Galois theory Differential field Dimension of a vector space Dimension of an algebraic variety Discrete logarithm Disquisitiones Arithmeticae Distributive property Distributivity Division (mathematics)Division by zero Division ring Domain (ring theory)Doubling the cube E. H. Moore Edmund F. Robertson Element (mathematics)Element (set theory)Elementarily equivalent Elliptic curve Elliptic curve cryptography Elliptic curves Emil Artin Encyclopedia of Mathematics Equation Equivalence of categories Ernst Kummer Ernst Steinitz Error correction code European Mathematical Society Existential quantifier Exponential field Ferdinand von Lindemann Fermat's Last Theorem Fernando Q. Gouvêa Field (disambiguation)Field (physics)Field automorphism Field extension Field homomorphism Field of fractions Field of rational functions Field with one element Finite extension Finite field First-order logic Formal power series Formally real field Fraction (mathematics)François Viète Frobenius map Frobenius theorem (real division algebras)Function (mathematics)Function field analogy Function field of an algebraic variety Fundamental theorem of Galois theory Fundamental theorem of algebra Galois cohomology Galois extension Galois group Galois representation Galois theory Game theory Gaussian elimination Gaussian rational Genus (mathematics)Geometric mean theorem German (language)Giuseppe Veronese Global field Glossary of field theory Graded ring Graduate Texts in Mathematics Grothendieck's Galois theory Group (mathematics)Group theory Group with operators Harald Niederreiter Hasse–Minkowski theorem Heinrich Martin Weber Heyting algebra Heyting field Holomorphic functions Hopf algebra Hyperreals Ideal (ring theory)Imaginary quadratic field Imaginary unit Indeterminate (variable)Infinitesimals Injective Injective map Integer Integral domain Integral domains Inverse Galois theory Invertible matrix Irrational number Irreducible polynomial Isomorphism Israel Kleiner (mathematician)J. W. S. Cassels Jacques Tits James Ax Jean-Pierre Serre John C. Baez John Horton Conway John Milnor Joseph-Louis Lagrange Joseph Liouville Journal für die reine und angewandte Mathematik Karl Georg Christian von Staudt Kronecker Jugendtraum Kronecker–Weber theorem Kurt Hensel Langlands program Lattice (order)Laurent series Lawrence C. Washington Lefschetz principle Leo Corry Leopold Kronecker Lie algebra Lie group Linear algebra Linear differential equation Linear equation over a ring Local field Local ring Local–global principle Lower bound Magma (algebra)Map of lattices Mathematical analysis Mathematical logic Mathematics Matsumoto's theorem (K-theory)Maximal ideal Meromorphic function Metric (mathematics)Michael Artin Michel Kervaire Midpoint Milnor K-theory Minimal model program Model theory Modular arithmetic Module (mathematics)Monoid Multiplication Multiplicative group Multiplicative identity Multiplicative inverse Nathan Jacobson Near-field (mathematics)Near-ring Nicolas Bourbaki Niels Henrik Abel Nimber Non-associative algebra Non-standard analysis Norm residue isomorphism theorem Normal extension Number field Number theory Octonion On Numbers and Games One-to-one correspondence Ordered field Ostrowski's theorem Otto Schreier P-adic analysis P-adic number P-adic valuation Paolo Ruffini (mathematician)Patrick Ion Paulo Ribenboim Perfectoid field Perpendicular Peter Gustav Lejeune Dirichlet Peter Scholze Pierre Deligne Plane (geometry)Polynomial Polynomial equation Polynomial ring Prentice Hall Prime ideal Prime number Primitive element theorem Principal ideal Profinite completion Proper class Proper map Prüfer group Puiseux series Quadratic form Quartic polynomial Quasifield Quasigroup Quaternion Quintic equation Quotient ring Racks and quandles Raoul Bott Rational fraction Rational number Real closed field Real number Regular function Regular pentagon Regular polygon Regular prime Residue class Residue field Richard Dedekind Riemann hypothesis Riemann zeta function Ring (mathematics)Ring homomorphism Ring theory Rng (algebra)Robert Fricke Root of a function Root of unity Scalar (mathematics)Scipione del Ferro Semicircle Semifield Semigroup Semilattice Semiring Separable extension Sequence Serge Lang Set (mathematics)Simple extension Smooth variety Solution in radicals Solvable group Splitting field Splitting of prime ideals in Galois extensions Springer-Verlag Square root Squaring the circle Straightedge Subgroup Subset Subtraction Surjective Surjective map Surreal number Symmetric group Symmetry group Tensor field Tensor product of fields Topological space Total order Transcendence basis Transcendence degree Transcendental extension Trivial ring Ultrafilter Ultrafilter lemma Ultraproduct Uncountable Uniformizer Unit (ring theory)Unramified extension Upper bound Vector field Vector space Vladimir Voevodsky Wedderburn's little theorem Weil conjectures Weil group Witt ring (forms)Zariski topology Zero of a function Étale fundamental group Évariste Galois Łoś's theorem

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