en.unionpedia.org

Field with one element, the Glossary

In mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist.^[1]

103 relations: Abc conjecture, Absorbing element, Abstract algebra, Abstract simplicial complex, Additive identity, Affine monoid, Affine space, Alain Connes, Aleksandr Smirnov, Algebraic curve, Algebraic geometry, Algebraic group, Algebraic K-theory, Algebraic number, Algebraic number theory, Algebraic variety, André Weil, Arakelov theory, Arithmetic derivative, Base (topology), Binomial coefficient, Building (mathematics), Category (mathematics), Caterina Consani, Characteristic (algebra), Christophe Soulé, Commutative algebra, Commutative ring, Complex geometry, Complex number, Computational complexity theory, Connected space, Cyclic group, Degeneracy (mathematics), Descent (mathematics), Diophantine equation, Discrete Fourier transform, Discrete Fourier transform over a ring, Dynkin diagram, Field extension, Finite field, Finite set, Flag (linear algebra), Flat morphism, Full and faithful functors, Function field of an algebraic variety, Functor, Grassmannian, Group action, Group Hopf algebra, ... Expand index (53 more) »
Abc conjecture
Noncommutative geometry

Abc conjecture

The abc conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory that arose out of a discussion of Joseph Oesterlé and David Masser in 1985.

See Field with one element and Abc conjecture

Absorbing element

In mathematics, an absorbing element (or annihilating element) is a special type of element of a set with respect to a binary operation on that set.

See Field with one element and Absorbing element

Abstract algebra

In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations acting on their elements.

See Field with one element and Abstract algebra

In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family.

See Field with one element and Abstract simplicial complex

Additive identity

In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element in the set, yields.

See Field with one element and Additive identity

Affine monoid

In abstract algebra, a branch of mathematics, an affine monoid is a commutative monoid that is finitely generated, and is isomorphic to a submonoid of a free abelian group \mathbb^d, d \ge 0.

See Field with one element and Affine monoid

Affine space

In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments.

See Field with one element and Affine space

Alain Connes

Alain Connes (born 1 April 1947 in Draguignan) is a French mathematician, known for his contributions to the study of operator algebras and noncommutative geometry.

See Field with one element and Alain Connes

Aleksandr Smirnov

Aleksandr or Alexander Smirnov may refer to.

See Field with one element and Aleksandr Smirnov

Algebraic curve

In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables.

See Field with one element and Algebraic curve

Algebraic geometry

Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems.

See Field with one element and Algebraic geometry

Algebraic group

In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety.

See Field with one element and Algebraic group

Algebraic K-theory

Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Field with one element and Algebraic K-theory are algebraic geometry.

See Field with one element and Algebraic K-theory

Algebraic number

An algebraic number is a number that is a root of a non-zero polynomial (of finite degree) in one variable with integer (or, equivalently, rational) coefficients.

See Field with one element and Algebraic number

Algebraic number theory

Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations.

See Field with one element and Algebraic number theory

Algebraic variety

Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Field with one element and algebraic variety are algebraic geometry.

See Field with one element and Algebraic variety

André Weil

André Weil (6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry.

See Field with one element and André Weil

Arakelov theory

In mathematics, Arakelov theory (or Arakelov geometry) is an approach to Diophantine geometry, named for Suren Arakelov. Field with one element and Arakelov theory are algebraic geometry.

See Field with one element and Arakelov theory

Arithmetic derivative

In number theory, the Lagarias arithmetic derivative or number derivative is a function defined for integers, based on prime factorization, by analogy with the product rule for the derivative of a function that is used in mathematical analysis.

See Field with one element and Arithmetic derivative

Base (topology)

In mathematics, a base (or basis;: bases) for the topology of a topological space is a family \mathcal of open subsets of such that every open set of the topology is equal to the union of some sub-family of \mathcal.

See Field with one element and Base (topology)

Binomial coefficient

In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem.

See Field with one element and Binomial coefficient

Building (mathematics)

In mathematics, a building (also Tits building, named after Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces.

See Field with one element and Building (mathematics)

Category (mathematics)

In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows".

See Field with one element and Category (mathematics)

Caterina Consani

Caterina (Katia) Consani (born 1963) is an Italian mathematician specializing in arithmetic geometry.

See Field with one element and Caterina Consani

Characteristic (algebra)

In mathematics, the characteristic of a ring, often denoted, is defined to be the smallest positive number of copies of the ring's multiplicative identity that will sum to the additive identity.

See Field with one element and Characteristic (algebra)

Christophe Soulé

Christophe Soulé (born 1951) is a French mathematician working in arithmetic geometry.

See Field with one element and Christophe Soulé

Commutative algebra

Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings.

See Field with one element and Commutative algebra

Commutative ring

In mathematics, a commutative ring is a ring in which the multiplication operation is commutative.

See Field with one element and Commutative ring

Complex geometry

In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. Field with one element and complex geometry are algebraic geometry.

See Field with one element and Complex geometry

Complex number

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted, called the imaginary unit and satisfying the equation i^.

See Field with one element and Complex number

Computational complexity theory

In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other.

See Field with one element and Computational complexity theory

Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets.

See Field with one element and Connected space

Cyclic group

In abstract algebra, a cyclic group or monogenous group is a group, denoted Cn (also frequently \Zn or Zn, not to be confused with the commutative ring of p-adic numbers), that is generated by a single element.

See Field with one element and Cyclic group

Degeneracy (mathematics)

In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class; "degeneracy" is the condition of being a degenerate case.

See Field with one element and Degeneracy (mathematics)

Descent (mathematics)

In mathematics, the idea of descent extends the intuitive idea of 'gluing' in topology. Field with one element and descent (mathematics) are algebraic geometry.

See Field with one element and Descent (mathematics)

Diophantine equation

In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, for which only integer solutions are of interest.

See Field with one element and Diophantine equation

Discrete Fourier transform

In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency.

See Field with one element and Discrete Fourier transform

Discrete Fourier transform over a ring

In mathematics, the discrete Fourier transform over a ring generalizes the discrete Fourier transform (DFT), of a function whose values are commonly complex numbers, over an arbitrary ring.

See Field with one element and Discrete Fourier transform over a ring

Dynkin diagram

In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line).

See Field with one element and Dynkin diagram

Field extension

In mathematics, particularly in algebra, a field extension (denoted L/K) is a pair of fields K \subseteq L, such that the operations of K are those of L restricted to K. In this case, L is an extension field of K and K is a subfield of L. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers.

See Field with one element and Field extension

Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. Field with one element and finite field are finite fields.

See Field with one element and Finite field

Finite set

In mathematics, particularly set theory, a finite set is a set that has a finite number of elements.

See Field with one element and Finite set

Flag (linear algebra)

In mathematics, particularly in linear algebra, a flag is an increasing sequence of subspaces of a finite-dimensional vector space V. Here "increasing" means each is a proper subspace of the next (see filtration): The term flag is motivated by a particular example resembling a flag: the zero point, a line, and a plane correspond to a nail, a staff, and a sheet of fabric.

See Field with one element and Flag (linear algebra)

Flat morphism

In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat map of rings, i.e., is a flat map for all P in X. A map of rings A\to B is called flat if it is a homomorphism that makes B a flat A-module.

See Field with one element and Flat morphism

Full and faithful functors

In category theory, a faithful functor is a functor that is injective on hom-sets, and a full functor is surjective on hom-sets.

See Field with one element and Full and faithful functors

Function field of an algebraic variety

In algebraic geometry, the function field of an algebraic variety V consists of objects that are interpreted as rational functions on V. In classical algebraic geometry they are ratios of polynomials; in complex geometry these are meromorphic functions and their higher-dimensional analogues; in modern algebraic geometry they are elements of some quotient ring's field of fractions.

See Field with one element and Function field of an algebraic variety

Functor

In mathematics, specifically category theory, a functor is a mapping between categories.

See Field with one element and Functor

Grassmannian

In mathematics, the Grassmannian \mathbf_k(V) (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all k-dimensional linear subspaces of an n-dimensional vector space V over a field K. For example, the Grassmannian \mathbf_1(V) is the space of lines through the origin in V, so it is the same as the projective space \mathbf(V) of one dimension lower than V. Field with one element and Grassmannian are algebraic geometry.

See Field with one element and Grassmannian

Group action

In mathematics, many sets of transformations form a group under function composition; for example, the rotations around a point in the plane.

See Field with one element and Group action

Group Hopf algebra

In mathematics, the group Hopf algebra of a given group is a certain construct related to the symmetries of group actions.

See Field with one element and Group Hopf algebra

Hasse–Weil zeta function

In mathematics, the Hasse–Weil zeta function attached to an algebraic variety V defined over an algebraic number field K is a meromorphic function on the complex plane defined in terms of the number of points on the variety after reducing modulo each prime number p. It is a global ''L''-function defined as an Euler product of local zeta functions. Field with one element and Hasse–Weil zeta function are algebraic geometry.

See Field with one element and Hasse–Weil zeta function

Homotopy groups of spheres

In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other.

See Field with one element and Homotopy groups of spheres

Hopf algebra

In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an (unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antihomomorphism satisfying a certain property.

See Field with one element and Hopf algebra

Identity element

In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. Field with one element and identity element are 1 (number).

See Field with one element and Identity element

Initial and terminal objects

In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in, there exists precisely one morphism.

See Field with one element and Initial and terminal objects

Institute of Combinatorics and its Applications

The Institute of Combinatorics and its Applications (ICA) is an international scientific organization formed in 1990 to increase the visibility and influence of the combinatorial community.

See Field with one element and Institute of Combinatorics and its Applications

Jacques Tits

Jacques Tits (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry.

See Field with one element and Jacques Tits

John C. Baez

John Carlos Baez (born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California.

See Field with one element and John C. Baez

Local zeta function

In number theory, the local zeta function (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as where is a non-singular -dimensional projective algebraic variety over the field with elements and is the number of points of defined over the finite field extension of. Field with one element and local zeta function are finite fields.

See Field with one element and Local zeta function

Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

See Field with one element and Mathematics

Matilde Marcolli

Matilde Marcolli is an Italian and American mathematical physicist.

See Field with one element and Matilde Marcolli

Mikhail Kapranov

Mikhail Kapranov, (Михаил Михайлович Капранов, born 1962) is a Russian mathematician, specializing in algebraic geometry, representation theory, mathematical physics, and category theory.

See Field with one element and Mikhail Kapranov

Module (mathematics)

In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring.

See Field with one element and Module (mathematics)

Monad (category theory)

In category theory, a branch of mathematics, a monad is a triple (T, \eta, \mu) consisting of a functor T from a category to itself and two natural transformations \eta, \mu that satisfy the conditions like associativity.

See Field with one element and Monad (category theory)

Monoid

In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element.

See Field with one element and Monoid

Multiplicative group

In mathematics and group theory, the term multiplicative group refers to one of the following concepts.

See Field with one element and Multiplicative group

Nikolai Durov

Nikolai Valeryevich Durov (Никола́й Вале́рьевич Ду́ров; born 21 November 1980) is a Russian programmer and mathematician.

See Field with one element and Nikolai Durov

Noncommutative geometry

Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions, possibly in some generalized sense. Field with one element and noncommutative geometry are algebraic geometry.

See Field with one element and Noncommutative geometry

Number theory

Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions.

See Field with one element and Number theory

Open set

In mathematics, an open set is a generalization of an open interval in the real line.

See Field with one element and Open set

Operation (mathematics)

In mathematics, an operation is a function which takes zero or more input values (also called "operands" or "arguments") to a well-defined output value.

See Field with one element and Operation (mathematics)

Pointed set

In mathematics, a pointed set (also based set or rooted set) is an ordered pair (X, x_0) where X is a set and x_0 is an element of X called the base point, also spelled basepoint.

See Field with one element and Pointed set

Polynomial ring

In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field.

See Field with one element and Polynomial ring

Prime ideal

In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers.

See Field with one element and Prime ideal

Projective geometry

In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations.

See Field with one element and Projective geometry

Projective space

In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity.

See Field with one element and Projective space

Q-Pochhammer symbol

In the mathematical field of combinatorics, the q-Pochhammer symbol, also called the q-shifted factorial, is the product (a;q)_n.

See Field with one element and Q-Pochhammer symbol

Quantum group

In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure.

See Field with one element and Quantum group

Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature.

See Field with one element and Real number

Riemann hypothesis

In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part.

See Field with one element and Riemann hypothesis

Riemann zeta function

The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s).

See Field with one element and Riemann zeta function

Riemann–Hurwitz formula

In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other.

See Field with one element and Riemann–Hurwitz formula

Root of unity

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power. Field with one element and root of unity are 1 (number).

See Field with one element and Root of unity

Scheme (mathematics)

In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x.

See Field with one element and Scheme (mathematics)

Schubert variety

In algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, \mathbf_k(V) of k-dimensional subspaces of a vector space V, usually with singular points. Field with one element and Schubert variety are algebraic geometry.

See Field with one element and Schubert variety

Semiring

In abstract algebra, a semiring is an algebraic structure.

See Field with one element and Semiring

Set (mathematics)

In mathematics, a set is a collection of different things; these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets.

See Field with one element and Set (mathematics)

Sheaf (mathematics)

In mathematics, a sheaf (sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them.

See Field with one element and Sheaf (mathematics)

Simplicial complex

In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration).

See Field with one element and Simplicial complex

Spectrum of a ring

In commutative algebra, the prime spectrum (or simply the spectrum) of a commutative ring R is the set of all prime ideals of R, and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the sheaf of rings \mathcal.

See Field with one element and Spectrum of a ring

Sphere spectrum

In stable homotopy theory, a branch of mathematics, the sphere spectrum S is the monoidal unit in the category of spectra.

See Field with one element and Sphere spectrum

Stable homotopy theory

In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor.

See Field with one element and Stable homotopy theory

Subcategory

In mathematics, specifically category theory, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in C with the same identities and composition of morphisms.

See Field with one element and Subcategory

Symmetric monoidal category

In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" \otimes is defined) such that the tensor product is symmetric (i.e. A\otimes B is, in a certain strict sense, naturally isomorphic to B\otimes A for all objects A and B of the category).

See Field with one element and Symmetric monoidal category

Toric variety

In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Field with one element and toric variety are algebraic geometry.

See Field with one element and Toric variety

Trivial semigroup

In mathematics, a trivial semigroup (a semigroup with one element) is a semigroup for which the cardinality of the underlying set is one.

See Field with one element and Trivial semigroup

Tropical geometry

In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication is replaced with ordinary addition: So for example, the classical polynomial x^3 + 2xy + y^4 would become \min\. Field with one element and tropical geometry are algebraic geometry.

See Field with one element and Tropical geometry

Unique games conjecture

In computational complexity theory, the unique games conjecture (often referred to as UGC) is a conjecture made by Subhash Khot in 2002.

See Field with one element and Unique games conjecture

Vector space

In mathematics and physics, 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''.

See Field with one element and Vector space

Weyl group

In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system.

See Field with one element and Weyl group

Yuri Manin

Yuri Ivanovich Manin (Ю́рий Ива́нович Ма́нин; 16 February 1937 – 7 January 2023) was a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics.

See Field with one element and Yuri Manin

Zariski topology

In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties.

See Field with one element and Zariski topology

Zero ring

In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element.

See Field with one element and Zero ring

References

[1] https://en.wikipedia.org/wiki/Field_with_one_element

Also known as Absolute mathematics, Field of characteristic one, Field of one element, Field with 1 element, Non-additive geometry, Nonadditive geometry.

, Hasse–Weil zeta function, Homotopy groups of spheres, Hopf algebra, Identity element, Initial and terminal objects, Institute of Combinatorics and its Applications, Jacques Tits, John C. Baez, Local zeta function, Mathematics, Matilde Marcolli, Mikhail Kapranov, Module (mathematics), Monad (category theory), Monoid, Multiplicative group, Nikolai Durov, Noncommutative geometry, Number theory, Open set, Operation (mathematics), Pointed set, Polynomial ring, Prime ideal, Projective geometry, Projective space, Q-Pochhammer symbol, Quantum group, Real number, Riemann hypothesis, Riemann zeta function, Riemann–Hurwitz formula, Root of unity, Scheme (mathematics), Schubert variety, Semiring, Set (mathematics), Sheaf (mathematics), Simplicial complex, Spectrum of a ring, Sphere spectrum, Stable homotopy theory, Subcategory, Symmetric monoidal category, Toric variety, Trivial semigroup, Tropical geometry, Unique games conjecture, Vector space, Weyl group, Yuri Manin, Zariski topology, Zero ring.

Field with one element, the Glossary

Table of Contents

See also

Abc conjecture

Noncommutative geometry

References