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, ..., Hamming space, Hasse principle, Hensel's lemma, Identity (mathematics), Integer, Irreducible polynomial, Isomorphism, Lagrange's theorem (group theory), Lenstra–Lenstra–Lovász lattice basis reduction algorithm, Linear algebra, Linear map, Linear subspace, Mathematics, Möbius function, Minimal polynomial (field theory), Modular arithmetic, Monic polynomial, Multiplicative group, National Institute of Standards and Technology, Number theory, Perfect field, Polynomial, Polynomial greatest common divisor, Polynomial ring, Prime number, Prime power, Primitive element (finite field), Product topology, Profinite group, Profinite integer, Quadratic residue, Quasi-finite field, Quotient ring, Rational number, Reciprocal polynomial, Root of unity, Set (mathematics), Splitting field, Trinomial, Union (set theory), Unique factorization domain, Vector space, Wedderburn's little theorem, Wiles's proof of Fermat's Last Theorem, Zech's logarithm, Zero of a function. Expand index (46 more) » « Shrink index
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.
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.
In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative.
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.
In mathematics, an automorphism is an isomorphism from a mathematical object to itself.
Évariste Galois (25 October 1811 – 31 May 1832) was a French mathematician.
In mathematics, any of the positive integers that occurs as a coefficient in the binomial theorem is a binomial coefficient.
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.
The Bulletin of the American Mathematical Society is a quarterly mathematical journal published by the American Mathematical Society.
Cambridge University Press (CUP) is the publishing business of the University of Cambridge.
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.
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 is the study of the properties of codes and their respective fitness for specific applications.
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.
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.
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 or cryptology (from κρυπτός|translit.
In algebra, a cyclic group or monogenous group is a group that is generated by a single element.
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 (DH)Synonyms of Diffie–Hellman key exchange include.
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.
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.
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.
In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible.
Eliakim Hastings Moore (January 26, 1862 – December 30, 1932), usually cited as E. H. Moore or E. Hastings Moore, was an American mathematician.
In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set.
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.
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 (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.
In arithmetic, Euclidean division is the process of division of two integers, which produces a quotient and a remainder smaller than the divisor.
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to.
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.
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.
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.
In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors.
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 states that if is a prime number, then for any integer, the number is an integer multiple of.
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.
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.
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.
A finite geometry is any geometric system that has only a finite number of points.
In abstract algebra, a finite group is a mathematical group with a finite number of elements.
In mathematics, more specifically abstract algebra, a finite ring is a ring that has a finite number of elements.
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.
The freshman's dream is a name sometimes given to the erroneous equation (x + y)n.
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.
In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function.
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.
In the field of algebra within mathematics, Galois theory, provides a connection between field theory and group theory.
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.
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.
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.
In mathematics an identity is an equality relation A.
An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").
In mathematics, an irreducible polynomial is, roughly speaking, a non-constant polynomial that cannot be factored into the product of two non-constant polynomials.
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, 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.
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 is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.
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.
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 (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
The classical Möbius function is an important multiplicative function in number theory and combinatorics.
In field theory, a branch of mathematics, the minimal polynomial of a value α is, roughly speaking, the polynomial of lowest degree having coefficients of a specified type, such that α is a root of the polynomial.
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli).
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.
In mathematics and group theory, the term multiplicative group refers to one of the following concepts.
The National Institute of Standards and Technology (NIST) is one of the oldest physical science laboratories in the United States.
Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.
In algebra, a field k is said to be perfect if any one of the following equivalent conditions holds.
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.
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.
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.
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
In mathematics, a prime power is a positive integer power of a single prime number.
In field theory, a primitive element of a finite field GF(q) is a generator of the multiplicative group of the field.
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.
In mathematics, profinite groups are topological groups that are in a certain sense assembled from finite groups.
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.
In mathematics, a quasi-finite field is a generalisation of a finite field.
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.
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.
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.
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.
In mathematics, a set is a collection of distinct objects, considered as an object in its own right.
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.
In elementary algebra, a trinomial is a polynomial consisting of three terms or monomials.
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.
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.
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.
In mathematics, Wedderburn's little theorem states that every finite domain is a field.
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 logarithms are used to implement addition in finite fields when elements are represented as powers of a generator \alpha.
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).