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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, an algebraic number is a complex number
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
 that is a root
Root (mathematics)

In mathematics, a root of a complex-valued Function is a member of the Domain of such that vanishes at , that is,In other words, a "root" of a function is a value for that produces a result of zero ....
 of a non-zero polynomial
Polynomial

In mathematics, a polynomial is an expression constructed from variables and constants, using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents....
 in one variable with rational
Rational number

In mathematics, a rational number is a number which can be expressed as a quotient of two integers. Non-integer rational numbers are usually written as the vulgar fraction , where b is not 0 ....
 (or equivalently, integer
Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
) coefficients. Complex numbers such as pi
Pi

Pi or p is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean geometry; this is the same value as the ratio of a circle's area to the square of its radius....
 that are not algebraic are said to be transcendental
Transcendental number

In mathematics, a transcendental number is a number that is not algebraic number, that is, not a solution of a non-zero polynomial equation with rational number coefficients....
, and are infinitely more numerous within the complex number field.


















sum, difference, product and quotient of two algebraic numbers is again algebraic, and the algebraic numbers therefore form a field
Field (mathematics)

In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms....
, sometimes denoted by (which may also denote the adele ring
Adele ring

In algebraic number theory and topological algebra, the adele ring is a topological ring which is built on the Field of rational numbers . It involves all the completions of the field....
) or .






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, an algebraic number is a complex number
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
 that is a root
Root (mathematics)

In mathematics, a root of a complex-valued Function is a member of the Domain of such that vanishes at , that is,In other words, a "root" of a function is a value for that produces a result of zero ....
 of a non-zero polynomial
Polynomial

In mathematics, a polynomial is an expression constructed from variables and constants, using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents....
 in one variable with rational
Rational number

In mathematics, a rational number is a number which can be expressed as a quotient of two integers. Non-integer rational numbers are usually written as the vulgar fraction , where b is not 0 ....
 (or equivalently, integer
Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
) coefficients. Complex numbers such as pi
Pi

Pi or p is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean geometry; this is the same value as the ratio of a circle's area to the square of its radius....
 that are not algebraic are said to be transcendental
Transcendental number

In mathematics, a transcendental number is a number that is not algebraic number, that is, not a solution of a non-zero polynomial equation with rational number coefficients....
, and are infinitely more numerous within the complex number field.

Examples

  • The rational number
    Rational number

    In mathematics, a rational number is a number which can be expressed as a quotient of two integers. Non-integer rational numbers are usually written as the vulgar fraction , where b is not 0 ....
    s, those expressed as the ratio of two whole numbers b and a, a not equal to zero, satisfy the above definition because x = −b/a is derived from (and satisfies) ax + b = 0. (In general, a or b can be negative, as can x).


  • Some irrational number
    Irrational number

    In mathematics, an irrational number is any real number that is not a rational number ? that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero....
    s are algebraic and some are not:
  • The numbers and are algebraic since they are the roots of and , respectively.


  • The golden ratio
    Golden ratio

    In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller....
     f is algebraic since it is a root of the polynomial .


  • The numbers π
    Pi

    Pi or p is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean geometry; this is the same value as the ratio of a circle's area to the square of its radius....
     and e
    E (mathematical constant)

    The mathematical constant e is the unique real number such that the function ex has the same value as the derivative, for all values of x....
     are not algebraic numbers (see the Lindemann–Weierstrass theorem
    Lindemann–Weierstrass theorem

    In mathematics, the Lindemann?Weierstrass theorem is a result that is very useful in establishing the transcendental number of numbers. It states that if α1, ..., αn are algebraic numbers which are linearly independent over the rational numbers Q, then eα1    ); hence they are transcendental.


  • The constructible number
    Constructible number

    A point in the Euclidean plane is a constructible point if, given a fixed coordinate system , the point can be constructed with Compass and straightedge constructions....
    s (those that, starting with a unit, can be constructed with straightedge and compass, e.g. the square root of 2) are algebraic.


  • The quadratic surds (roots of a quadratic equation with integral coefficents a, b, and c) are algebraic numbers. Thus those complex numbers derived from — those corresponding to the case when the exponent — are called quadratic numbers. If the quadratic equation is monic (a = 1) then the roots are quadratic integer
    Quadratic integer

    In mathematics, quadratic integers are the integral solutions of the equations of the form:for some integers B and C. Such integers have the form...
    s.


  • Gaussian integer
    Gaussian integer

    A Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]....
    s — those complex numbers a + bi where both a and b are integers are also quadratic integers.


  • When the lead coefficient of a polynomial is 1, any value x that satisfies the polynomial is said to be an algebraic integer
    Algebraic integer

    This article deals with the ring of complex numbers integral over Z. For the general notion of algebraic integer, see Integrality.In number theory, an algebraic integer is a complex number that is a root of some monic polynomial with integer coefficients....
    . Note that an "algebraic integer" need not be a counting number such as 1, 2, 3, ... or a negative counterpart.


  • This definition comes from the notion that satisfies , and when then (i.e. b here being a positive or negative counting number or 0). But observe that from , and . So these two x are "algebraic integers" as well. This applies for any value of lead-exponent n. (See more below).

Properties

  • The set of algebraic numbers is countable
    Countable set

    In mathematics, a countable set is a Set with the same cardinality as some subset of the set of natural numbers. The term was originated by Georg Cantor; it stems from the fact that the natural numbers are often called counting numbers....
     (enumerable).
  • Hence, the set of algebraic numbers has Lebesgue measure
    Lebesgue measure

    In mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a length, area or volume to subsets of Euclidean space....
     zero (as a subset of the complex numbers), i.e. "almost all
    Almost everywhere

    In measure theory , one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i.e....
    " complex numbers are not algebraic.
  • Given an algebraic number, there is a unique monic polynomial (with rational coefficients) of least degree
    Polynomial

    In mathematics, a polynomial is an expression constructed from variables and constants, using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents....
     that has the number as a root. This polynomial is called its minimal polynomial
    Minimal polynomial (field theory)

    In field theory , given a field extension E / F and an element α of E which is an algebraic element over F, the minimal polynomial of α is the polynomial p, with coefficients in F, of least degree such that p = 0....
    . If its minimal polynomial has degree n, then the algebraic number is said to be of degree n. An algebraic number of degree 1 is a rational number
    Rational number

    In mathematics, a rational number is a number which can be expressed as a quotient of two integers. Non-integer rational numbers are usually written as the vulgar fraction , where b is not 0 ....
    .
  • All algebraic numbers are computable
    Computable number

    In mathematics, theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers or the computable reals, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm....
     and therefore definable
    Definable number

    A real number a is first-order definable in the language of set theory, without parameters, if there is a formula f in the language of set theory, with one free variable, such that a is the unique real number such that f holds ....
    .


The field of algebraic numbers

The sum, difference, product and quotient of two algebraic numbers is again algebraic, and the algebraic numbers therefore form a field
Field (mathematics)

In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms....
, sometimes denoted by (which may also denote the adele ring
Adele ring

In algebraic number theory and topological algebra, the adele ring is a topological ring which is built on the Field of rational numbers . It involves all the completions of the field....
) or . Every root of a polynomial equation whose coefficients are algebraic numbers is again algebraic. This can be rephrased by saying that the field of algebraic numbers is algebraically closed
Algebraically closed field

In mathematics, a field F is said to be algebraically closed if every polynomial in one variable of degree at least 1, with coefficients in F, has a root in F....
. In fact, it is the smallest algebraically closed field containing the rationals, and is therefore called the algebraic closure
Algebraic closure

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

Numbers defined by radicals

All numbers which can be obtained from the integers using a finite
Finite set

In mathematics, finite set is a Set that has a finite number of element . For example,is a finite set with five elements. The number of elements of a finite set is a natural number , and is called the cardinality of the set....
 number of addition
Addition

Addition is the mathematics process of putting things together. The plus sign "+" means that numbers are added together. For example, in the picture on the right, there are 3 + 2 apples?meaning three apples and two other apples?which is the same as five apples, since 3 + 2 = 5....
s, subtraction
Subtraction

Subtraction is one of the four basic arithmetic operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with....
s, multiplication
Multiplication

Multiplication is the Operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic .Multiplication is defined for Natural number in terms of repeated addition; for example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together:...
s, division
Division (mathematics)

In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication.Specifically, if c times b equals a, written:...
s, and taking nth roots (where n is a positive integer) are algebraic. The converse, however, is not true: there are algebraic numbers which cannot be obtained in this manner. All of these numbers are solutions to polynomials of degree = 5. This is a result of Galois theory
Galois theory

In mathematics, more specifically in abstract algebra, Galois theory, named after ?variste Galois, provides a connection between field theory and group theory....
 (see Quintic equation
Quintic equation

In mathematics, a quintic equation is a polynomial equation of Degree of a polynomial five. It is of the form:where .......
s and the Abel–Ruffini theorem
Abel–Ruffini theorem

The Abel?Ruffini theorem states that there is no general solution in Radical to polynomial equations of degree five or higher....
). An example of such a number is the unique real root of (which is approximately 1.167303978261418684256).

Algebraic integers

An algebraic integer
Algebraic integer

This article deals with the ring of complex numbers integral over Z. For the general notion of algebraic integer, see Integrality.In number theory, an algebraic integer is a complex number that is a root of some monic polynomial with integer coefficients....
 is a number which is a root of a polynomial with integer coefficients (that is, an algebraic number) with leading coefficient 1 (a monic polynomial). Examples of algebraic integers are 3v + 5, 6i − 2 and (1 + iv)/2.

The sum, difference and product of algebraic integers are again algebraic integers, which means that the algebraic integers form a ring. The name algebraic integer comes from the fact that the only rational numbers which are algebraic integers are the integers, and because the algebraic integers in any number field
Algebraic number field

In mathematics, an algebraic number field F is a finite, field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite Hamel dimension, when considered as a vector space over Q....
 are in many ways analogous to the integers. If K is a number field, its ring of integers
Ring of integers

In mathematics, the ring of integers is the Set of integers made an algebraic structure Z with the operations of integer addition, negation, and multiplication....
 is the subring of algebraic integers in K, and is frequently denoted as OK. These are the prototypical examples of Dedekind domain
Dedekind domain

In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product...
s.

Special classes of algebraic number

  • Gaussian integer
    Gaussian integer

    A Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]....
  • Eisenstein integer
    Eisenstein integer

    In mathematics, Eisenstein integers, named after Ferdinand Eisenstein, are complex numbers of the formwhere a and b are integers and...
  • Quadratic irrational
    Quadratic irrational

    In mathematics, a quadratic irrational, also known asa quadratic irrationality or quadratic surd, is an irrational number that is the solution to some quadratic equation with rational coefficients....
  • Fundamental unit
    Fundamental unit (number theory)

    In algebraic number theory, a fundamental unit is a generator for the Torsion -free unit group of the Algebraic integer of a number field, when that group is infinite cyclic....
  • Root of unity
    Root of unity

    In mathematics, the nth roots of unity, or Abraham de Moivre numbers, are all the complex numbers that yield 1 when exponentiation to a given power n....
  • Gaussian period
    Gaussian period

    In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of root of unity. They permit explicit calculations in cyclotomic fields, in relation both with Galois theory and with harmonic analysis ....
  • Pisot-Vijayaraghavan number
    Pisot-Vijayaraghavan number

    In mathematics, a Pisot-Vijayaraghavan number, also called simply a Pisot number or a PV number, is an algebraic integer α which is real and exceeds 1, but such that its conjugate elements are all less than 1 in absolute value or modulus....
  • Salem number
    Salem number

    In mathematics, a real_number algebraic integer α > 1 is a Salem number if all its conjugate roots have absolute value no greater than 1, and at least one has absolute value exactly 1....