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Commutative property
The commutative property, or commutative law, is the property of a binary operation that swapping the order of its two inputs never changes the result.
Commutative ring
A commutative ring is a ring in which multiplication is commutative, meaning a·b = b·a for all elements; examples include the integers, the rationals, and the real numbers.
Commutator
A commutator is a group element of the form [g,h] = g⁻¹h⁻¹gh that equals the identity exactly when g and h commute, measuring noncommutativity in group theory.
Commutator subgroup
The commutator subgroup, also called the derived subgroup, of a group is the subgroup generated by all commutators; it is the unique smallest normal subgroup whose quotient is abelian.
Comodule
A comodule, also called a corepresentation, is a vector space with a coaction map from a coalgebra, the arrow-reversed analog of a module over an algebra.
Comonad
A comonad is a category-theory structure dual to a monad: an endofunctor with a counit and comultiplication, arising from adjunctions and used in sheaf cohomology.
Companion matrix
A companion matrix is an n × n matrix whose characteristic and minimal polynomial equal a given monic polynomial, used to find polynomial roots as eigenvalues.
Complete Boolean algebra
A complete Boolean algebra, also called a complete Boolean lattice, is a Boolean algebra in which every subset, even an infinite one, has a join and a meet.
Complete category
A complete category is a category in which every small diagram has a limit, so products, equalizers, pullbacks, and terminal objects all exist; examples include Set, Top, and Group.
Complete lattice
A complete lattice is a partially ordered set in which every subset has a least upper bound (join) and a greatest lower bound (meet) within the set.
Completely bounded and completely positive maps
Completely bounded maps are linear maps between operator algebras whose norm stays uniformly bounded on matrices of every size; completely positive maps are positive at every matrix level.
Completeness of the real numbers
Completeness of the real numbers, also called the completeness axiom, is the property that every nonempty set of reals bounded above has a least upper bound.
Completing the square
Completing the square is an algebraic technique that rewrites a quadratic expression as a squared term plus a constant, used to solve quadratic equations and find a parabola's vertex.
Complex conjugate
The complex conjugate, or complex conjugation, maps a complex number x + iy to x − iy, reflecting it across the real axis in the complex plane.
Complex multiplication
Complex multiplication, or CM, is a property of an elliptic curve whose endomorphism ring is larger than the integers, exemplified by the curve y² = x³ + x.
Complex multiplication of abelian varieties
Complex multiplication of abelian varieties is the property that an abelian variety's endomorphism algebra is as large as possible, developed by Shimura and Taniyama and underlying class field theory.
Complex number
A complex number is a number x + iy, where x and y are real and i² = −1, completing the real numbers into an algebraically closed field.
Complex plane
The complex plane, also called the Argand plane, represents complex numbers as points, with the real part on the horizontal axis and the imaginary part on the vertical.
Complexity of numerical linear algebra
The complexity of numerical linear algebra counts the arithmetic operations needed for core matrix problems such as solving linear systems, least squares, inversion, and eigenvalue computation.
Composite number
A composite number, also called a factorable number, is a positive integer greater than 1 that is not prime, such as 6 = 2 × 3.
Composition algebra
A composition algebra is a not-necessarily-associative algebra with a nondegenerate quadratic form whose norm multiplies over products, with the real numbers, complex numbers, quaternions, and octonions as classical examples.
Composition series
A composition series of a group is a finite chain of subgroups, each normal in the next, whose simple quotients are its composition factors, unique by the Jordan–Hölder theorem.
Compressed sensing
Compressed sensing (also called compressive sampling) is a signal acquisition technique, formalized around 2004, that recovers sparse signals from far fewer measurements than classical sampling theory requires.
Computational algebraic number theory
Computational algebraic number theory is the study of algorithms for computing with algebraic number fields, framed by Hendrik Lenstra's 1992 survey around Galois groups, rings of integers, and class groups.
Computational complexity of matrix multiplication
The computational complexity of matrix multiplication is measured by the exponent ω: Strassen's 1969 algorithm gave ω < 2.81, and the best bound is now ω < 2.371177.
Computational group theory
Computational group theory (CGT) is the study of algorithms for groups, ranging from provably unsolvable decision problems to practical systems such as GAP and Magma.
Computer algebra
Computer algebra, also called symbolic computation, is the branch of computer science that develops software for the exact, symbolic solution of equations, unlike approximate numerical computing.
Conductor (class field theory)
In class field theory, the conductor of a finite abelian extension, also called the conductor ideal, is the smallest modulus whose ray class field contains the extension.
Conformal algebra
A conformal algebra is the Lie algebra of infinitesimal conformal transformations of a geometry or field theory, central to conformal field theory and the operator product expansion.
Congruence lattice
The congruence lattice of an algebra is the lattice of all its congruence relations ordered by inclusion, generalizing normal subgroups and ideals; every algebraic lattice arises this way.