Field (mathematics)Algebraic structure with addition, multiplication, and divisionA field is an algebraic structure that is closed under the four usual arithmetic operations (except when it comes to division by zero).In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational numbers do.
Complex numberNumber with a real and an imaginary partA complex number z = a + bi can be visually represented as a pair of numbers (a, b) forming a position vector (blue) or a point (red) on an Argand diagram, representing the complex plane.
Real numberIn mathematics, a real number is a number that can be used to measure a continuous one-dimensionalquantity such as a length, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion.
Algebraic number theoryAlgebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields.
Galois theoryIn mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand. Galois introduced the subject for studying roots of polynomials.
AlgebraAlgebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication. Elementary algebra is the main form of algebra taught in schools.
Restriction (mathematics)In mathematics, the restriction of a functionf{\displaystyle f} is a new function, denoted f|A{\displaystyle f\vert _{A}} or f↾A,{\displaystyle f{\upharpoonright _{A}},} obtained by choosing a smaller domainA{\displaystyle A} for the original function f.{\displaystyle f.} The function f{\displaystyle f} is then said to extendf|A.{\displaystyle f\vert _{A}.}Formal definitionLet f:E→F{\displaystyle f:E\to F} be a function from a setE{\displaystyle E} to a set F.{\displaystyle F.} If a set A{\displaystyle A} is a subset of E,{\displaystyle E,} then the restriction of f{\displaystyle f}to A{\displaystyle A} is the function f|A:A→F{\displaystyle {f|}_{A}:A\to F} given by f|A(x)=f(x){\displaystyle {f|}_{A}(x)=f(x)} for x∈A.{\displaystyle x\in A.} Informally, the restriction of f{\displaystyle f} to A{\displaystyle A} is the same function as f,{\displaystyle f,} but is only defined on A{\displaystyle A}.
MultiplicationArithmetical operationFour bags with three marbles per bag gives twelve marbles (4 × 3 = 12).Multiplication can also be thought of as scaling. Here, 2 is being multiplied by 3 using scaling, giving 6 as a result.Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division. The result of a multiplication operation is called a product.
AdditionAddition, usually denoted with the plus sign +, is one of the four basic operations of arithmetic, the other three being subtraction, multiplication, and division. The addition of two whole numbers results in the total or sum of those values combined. For example, the adjacent image shows two columns of apples, one with three apples and the other with two apples, totaling to five apples.Algebraic geometryBranch of mathematics This Togliatti surface is an algebraic surface of degree five. The picture represents a portion of its real locus. Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems. Classically, it studies zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects.
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.
Zero of a functionIn mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued functionf{\displaystyle f}, is a member x{\displaystyle x} of the domain of f{\displaystyle f} such that f(x){\displaystyle f(x)}vanishes at x{\displaystyle x}; that is, the function f{\displaystyle f} attains the value of 0 at x{\displaystyle x}, or equivalently, x{\displaystyle x} is a solution to the equation f(x)=0{\displaystyle f(x)=0}.
field extensionRelated word or term.extension fieldRelated word or term.*As an Amazon Associate I earn from qualifying purchases.