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# 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. [1]

96 relations: Abelian group, Algebraic closure, Algebraic geometry, Alternative algebra, Artin–Zorn theorem, Automorphism, Évariste Galois, Binomial coefficient, Binomial theorem, Bulletin of the American Mathematical Society, Cambridge University Press, Characteristic (algebra), Chinese remainder theorem, Coding theory, Computer algebra system, Coprime integers, Cryptographic protocol, Cryptography, Cyclic group, Cyclotomic polynomial, Diffie–Hellman key exchange, Direct limit, Discrete logarithm, Discriminant, Division ring, E. H. Moore, Element (mathematics), Elementary abelian group, Elliptic curve, Elliptic-curve Diffie–Hellman, Euclidean division, Euler's totient function, Exponentiation by squaring, Expression (mathematics), Factorization of polynomials, Factorization of polynomials over finite fields, Ferdinand Georg Frobenius, Fermat's little theorem, Field (mathematics), Field with one element, Finite field arithmetic, Finite geometry, Finite group, Finite ring, Formal derivative, Freshman's dream, Frobenius endomorphism, Function composition, Galois extension, Galois theory, ... Expand index (46 more) »

## Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

## Algebraic closure

In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.

## Algebraic geometry

Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.

## Alternative algebra

In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative.

## Artin–Zorn theorem

In mathematics, the Artin–Zorn theorem, named after Emil Artin and Max Zorn, states that any finite alternative division ring is necessarily a finite field.

## Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself.

## Évariste Galois

Évariste Galois (25 October 1811 – 31 May 1832) was a French mathematician.

## Binomial coefficient

In mathematics, any of the positive integers that occurs as a coefficient in the binomial theorem is a binomial coefficient.

## Binomial theorem

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.

## Bulletin of the American Mathematical Society

The Bulletin of the American Mathematical Society is a quarterly mathematical journal published by the American Mathematical Society.

## Cambridge University Press

Cambridge University Press (CUP) is the publishing business of the University of Cambridge.

## Characteristic (algebra)

In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0) if the sum does indeed eventually attain 0.

## Chinese remainder theorem

The Chinese remainder theorem is a theorem of number theory, which states that if one knows the remainders of the Euclidean division of an integer by several integers, then one can determine uniquely the remainder of the division of by the product of these integers, under the condition that the divisors are pairwise coprime.

## Coding theory

Coding theory is the study of the properties of codes and their respective fitness for specific applications.

## Computer algebra system

A computer algebra system (CAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists.

## Coprime integers

In number theory, two integers and are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1.

## Cryptographic protocol

A security protocol (cryptographic protocol or encryption protocol) is an abstract or concrete protocol that performs a security-related function and applies cryptographic methods, often as sequences of cryptographic primitives.

## Cryptography

Cryptography or cryptology (from κρυπτός|translit.

## Cyclic group

In algebra, a cyclic group or monogenous group is a group that is generated by a single element.

## Cyclotomic polynomial

In mathematics, more specifically in algebra, the nth cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any.

## Diffie–Hellman key exchange

Diffie–Hellman key exchange (DH)Synonyms of Diffie–Hellman key exchange include.

## Direct limit

In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way.

## Discrete logarithm

In the mathematics of the real numbers, the logarithm logb a is a number x such that, for given numbers a and b. Analogously, in any group G, powers bk can be defined for all integers k, and the discrete logarithm logb a is an integer k such that.

## Discriminant

In algebra, the discriminant of a polynomial is a polynomial function of its coefficients, which allows deducing some properties of the roots without computing them.

## Division ring

In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible.

## E. H. Moore

Eliakim Hastings Moore (January 26, 1862 – December 30, 1932), usually cited as E. H. Moore or E. Hastings Moore, was an American mathematician.

## Element (mathematics)

In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set.

## Elementary abelian group

In group theory, an elementary abelian group (or elementary abelian p-group) is an abelian group in which every nontrivial element has order p. The number p must be prime, and the elementary abelian groups are a particular kind of ''p''-group.

## Elliptic curve

In mathematics, an elliptic curve is a plane algebraic curve defined by an equation of the form which is non-singular; that is, the curve has no cusps or self-intersections.

## Elliptic-curve Diffie–Hellman

Elliptic-curve Diffie–Hellman (ECDH) is an anonymous key agreement protocol that allows two parties, each having an elliptic-curve public–private key pair, to establish a shared secret over an insecure channel.

## Euclidean division

In arithmetic, Euclidean division is the process of division of two integers, which produces a quotient and a remainder smaller than the divisor.

## Euler's totient function

In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to.

## Exponentiation by squaring

In mathematics and computer programming, exponentiating by squaring is a general method for fast computation of large positive integer powers of a number, or more generally of an element of a semigroup, like a polynomial or a square matrix.

## Expression (mathematics)

In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.

## Factorization of polynomials

In mathematics and computer algebra, factorization of polynomials or polynomial factorization is the process of expressing a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain.

## Factorization of polynomials over finite fields

In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors.

## Ferdinand Georg Frobenius

Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory.

## Fermat's little theorem

Fermat's little theorem states that if is a prime number, then for any integer, the number is an integer multiple of.

## Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

## Field with one element

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.

## Finite field arithmetic

In mathematics, finite field arithmetic is arithmetic in a finite field (a field containing a finite number of elements) as opposed to arithmetic in a field with an infinite number of elements, like the field of rational numbers.

## Finite geometry

A finite geometry is any geometric system that has only a finite number of points.

## Finite group

In abstract algebra, a finite group is a mathematical group with a finite number of elements.

## Finite ring

In mathematics, more specifically abstract algebra, a finite ring is a ring that has a finite number of elements.

## Formal derivative

In mathematics, the formal derivative is an operation on elements of a polynomial ring or a ring of formal power series that mimics the form of the derivative from calculus.

## Freshman's dream

The freshman's dream is a name sometimes given to the erroneous equation (x + y)n.

## Frobenius endomorphism

In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic, an important class which includes finite fields.

## Function composition

In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function.

## Galois extension

In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.

## Galois theory

In the field of algebra within mathematics, Galois theory, provides a connection between field theory and group theory.

## Hamming space

In statistics and coding theory, a Hamming space is usually the set of all 2^N binary strings of length N. It is used in the theory of coding signals and transmission.

## Hasse principle

In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number.

## Hensel's lemma

In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a polynomial equation has a simple root modulo a prime number, then this root corresponds to a unique root of the same equation modulo any higher power of, which can be found by iteratively "lifting" the solution modulo successive powers of.

## Identity (mathematics)

In mathematics an identity is an equality relation A.

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## Irreducible polynomial

In mathematics, an irreducible polynomial is, roughly speaking, a non-constant polynomial that cannot be factored into the product of two non-constant polynomials.

## Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.

## Lagrange's theorem (group theory)

Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order (number of elements) of every subgroup H of G divides the order of G. The theorem is named after Joseph-Louis Lagrange.

## Lenstra–Lenstra–Lovász lattice basis reduction algorithm

The Lenstra–Lenstra–Lovász (LLL) lattice basis reduction algorithm is a polynomial time lattice reduction algorithm invented by Arjen Lenstra, Hendrik Lenstra and László Lovász in 1982.

## Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.

## Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

## Linear subspace

In linear algebra and related fields of mathematics, a linear subspace, also known as a vector subspace, or, in the older literature, a linear manifold, is a vector space that is a subset of some other (higher-dimension) vector space.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Möbius function

The classical Möbius function is an important multiplicative function in number theory and combinatorics.

## Minimal polynomial (field theory)

In field theory, a branch of mathematics, the minimal polynomial of a value &alpha; is, roughly speaking, the polynomial of lowest degree having coefficients of a specified type, such that &alpha; is a root of the polynomial.

## Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli).

## Monic polynomial

In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1.

## Multiplicative group

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

## National Institute of Standards and Technology

The National Institute of Standards and Technology (NIST) is one of the oldest physical science laboratories in the United States.

## Number theory

Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.

## Perfect field

In algebra, a field k is said to be perfect if any one of the following equivalent conditions holds.

## Polynomial

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

## Polynomial greatest common divisor

In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials.

## Polynomial ring

In mathematics, especially in the field of abstract 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.

## Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

## Prime power

In mathematics, a prime power is a positive integer power of a single prime number.

## Primitive element (finite field)

In field theory, a primitive element of a finite field GF(q) is a generator of the multiplicative group of the field.

## Product topology

In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology.

## Profinite group

In mathematics, profinite groups are topological groups that are in a certain sense assembled from finite groups.

## Profinite integer

In mathematics, a profinite integer is an element of the ring where p runs over all prime numbers, \mathbb_p is the ring of ''p''-adic integers and \widehat.

In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that: Otherwise, q is called a quadratic nonresidue modulo n. Originally an abstract mathematical concept from the branch of number theory known as modular arithmetic, quadratic residues are now used in applications ranging from acoustical engineering to cryptography and the factoring of large numbers.

## Quasi-finite field

In mathematics, a quasi-finite field is a generalisation of a finite field.

## Quotient ring

In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient groups of group theory and the quotient spaces of linear algebra.

## Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

## Reciprocal polynomial

In algebra, the reciprocal polynomial, or reflected polynomial* or, of a polynomial of degree with coefficients from an arbitrary field, such as is the polynomial Essentially, the coefficients are written in reverse order.

## Root of unity

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power.

## Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

## Splitting field

In abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of that field over which the polynomial splits or decomposes into linear factors.

## Trinomial

In elementary algebra, a trinomial is a polynomial consisting of three terms or monomials.

## Union (set theory)

In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.

## Unique factorization domain

In mathematics, a unique factorization domain (UFD) is an integral domain (a non-zero commutative ring in which the product of non-zero elements is non-zero) in which every non-zero non-unit element can be written as a product of prime elements (or irreducible elements), uniquely up to order and units, analogous to the fundamental theorem of arithmetic for the integers.

## Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

## Wedderburn's little theorem

In mathematics, Wedderburn's little theorem states that every finite domain is a field.

## Wiles's proof of Fermat's Last Theorem

Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves.

## Zech's logarithm

Zech logarithms are used to implement addition in finite fields when elements are represented as powers of a generator \alpha.

## Zero of a function

In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) vanishes at x; that is, x is a solution of the equation f(x).

## References

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