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Étale cohomology
Étale cohomology is a cohomology theory for algebraic varieties and schemes in which étale morphisms act as covers, developed by Grothendieck and Artin to prove the Weil conjectures, completed by Deligne in 1974.
Étale fundamental group
The étale fundamental group, also called the algebraic fundamental group, is an invariant of a scheme introduced by Alexander Grothendieck that classifies its finite étale covers as the topological fundamental group classifies covering spaces.
Étale morphism
An étale morphism, or étale map, is a morphism of schemes that is smooth of relative dimension 0, the analog of a local diffeomorphism or covering map in topology.
Euclid's lemma
Euclid's lemma, also called Euclid's first theorem, states that a prime dividing a product of two integers must divide one factor, and it proves unique prime factorization.
Euclid's theorem
Euclid's theorem, on the infinitude of primes, states that there are infinitely many prime numbers; Euclid proved it around 300 BC as Proposition 20 of Book IX of the Elements.
Euclidean algorithm
The Euclidean algorithm, also called Euclid's algorithm, is a procedure for finding the greatest common divisor of two integers, described by Euclid around 300 BC.
Euclidean division
Euclidean division, also called division with remainder, is the theorem that dividing one integer by a nonzero integer yields a unique quotient and remainder smaller than the divisor.
Euclidean domain
A Euclidean domain is an integral domain with a Euclidean norm allowing division with remainder, as in the integers, so the Euclidean algorithm computes greatest common divisors.
Euler numbers
The Euler numbers are a sequence of integers introduced by Leonhard Euler in 1755, defined as the coefficients of the hyperbolic secant, beginning 1, -1, 5, -61, 1385.
Euler product
An Euler product is a representation of a Dirichlet series or L-function as a convergent infinite product, one factor per prime, introduced by Euler in 1737.
Euler's identity
Euler's identity, also written e^iπ = −1, is the mathematical statement that e to the power i times pi equals negative 1, connecting five fundamental constants.
Euler's sum of powers conjecture
Euler's sum of powers conjecture, also called the Euler quartic conjecture, is a disproved 1769 conjecture of Leonhard Euler that at least n nth powers are needed to sum to an nth power.
Euler's theorem
Euler's theorem, also called Euler's totient theorem, is a number theory result stating that a to the power phi(n) equals 1 modulo n when a and n are coprime.
Euler's totient function
Euler's totient function, also called Euler's phi function, counts the positive integers up to n coprime to n, and it is central to Euler's theorem and RSA encryption.
Évariste Galois
Évariste Galois (1811–1832) was a French mathematician who founded Galois theory and gave the precise criterion for solving polynomial equations by radicals; he died in a duel at 20.
Exact sequence
An exact sequence is a sequence of algebraic objects, typically modules, connected by homomorphisms in which each map's image equals the next map's kernel, the central notion of homological algebra.
Exchange matrix
The exchange matrix J is the square matrix with ones on the antidiagonal and zeros elsewhere; multiplying by it reverses a matrix's rows or columns.
Explicit formulae (L-function)
An explicit formula for an L-function is an identity in analytic number theory expressing a sum over the L-function's nontrivial zeros as a sum over prime powers.
Exploratory factor analysis
Exploratory factor analysis (EFA) is a multivariate statistical method that models correlations among observed variables as the shared influence of a smaller number of unobserved common factors, without imposing a factor structure in advance.
Exponential sum
An exponential sum, also called a trigonometric sum, is a finite sum of complex roots of unity, central to analytic number theory since Gauss's 1801 work.
Exponentiation
Exponentiation is the arithmetic operation that raises a base to an exponent, written as a superscript; for positive integer exponents it means repeated multiplication, extending to zero, negative, rational, real, and complex exponents.
Expression (mathematics)
In mathematics, an expression is a finite combination of numbers, variables, and operation symbols standing for a calculation, and an equation joins two expressions with an equal sign.
Ext functor
The Ext functor, or Ext group, is a family of functors in homological algebra defined as the right derived functors of the Hom functor, named for extensions.
Extended Euclidean algorithm
The extended Euclidean algorithm, also called extended GCD, computes the greatest common divisor of two integers plus Bézout coefficients, and is used to find modular inverses in cryptography.
Extended precision
Extended precision is an IEEE 754 floating-point format exceeding its associated basic format in precision and range, in practice usually Intel's x87 80-bit format with 64 significant bits.
Extendible cardinal
An extendible cardinal is a cardinal κ in set theory such that every rank segment starting at κ can be elementarily copied into a higher one.
Exterior algebra
Exterior algebra, also called Grassmann algebra, is the algebra of a vector space in which vectors anticommute under the wedge product. It packages oriented volumes and determinants.
Exterior power
The k-th exterior power ΛᵏV of a vector space or module V is built from the k-fold tensor power by forcing tensors with a repeated factor to vanish.
F-algebra
An F-algebra, or algebra for an endofunctor, is a category-theory construction generalizing algebraic structure: a pair (A, α) with α : F(A) → A for an endofunctor F.
F4 Lie algebra
F4 is one of the five exceptional simple Lie algebras, a 52-dimensional complex simple Lie algebra of rank 4 whose root system has two root lengths, classified by Killing and Cartan.