The square root of 2 is an algebraic number equal to the length of the hypotenuse of a right triangle with legs of length 1.

An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, , is an algebraic number, because it is a root of the polynomial x2x − 1. That is, it is a value for x for which the polynomial evaluates to zero. As another example, the complex number is algebraic because it is a root of x4 + 4.

All integers and rational numbers are algebraic, as are all roots of integers. Real and complex numbers that are not algebraic, such as π and e, are called transcendental numbers.

The set of algebraic numbers is countably infinite and has measure zero in the Lebesgue measure as a subset of the uncountable complex numbers. In that sense, almost all complex numbers are transcendental.



Algebraic numbers on the complex plane colored by degree (bright orange/red = 1, green = 2, blue = 3, yellow = 4)


Algebraic numbers colored by degree (blue = 4, cyan = 3, red = 2, green = 1). The unit circle is black.

The sum, difference, product, and quotient (if the denominator is nonzero) of two algebraic numbers is again algebraic, as can be demonstrated by using the resultant, and algebraic numbers thus form a field[7] (sometimes denoted by , but that usually denotes the adele ring). Every root of a polynomial equation whose coefficients are algebraic numbers is again algebraic. That can be rephrased by saying that the field of algebraic numbers is algebraically closed. In fact, it is the smallest algebraically-closed field containing the rationals and so it is called the algebraic closure of the rationals.

Related fields

Numbers defined by radicals

Any number that can be obtained from the integers using a finite number of additions, subtractions, multiplications, divisions, and taking (possibly complex) nth roots where n is a positive integer are algebraic. The converse, however, is not true: there are algebraic numbers that cannot be obtained in this manner. These numbers are roots of polynomials of degree 5 or higher, a result of Galois theory (see Quintic equations and the Abel–Ruffini theorem). For example, the equation:

has a unique real root that cannot be expressed in terms of only radicals and arithmetic operations.

Closed-form number

Main article: Closed-form number

Algebraic numbers are all numbers that can be defined explicitly or implicitly in terms of polynomials, starting from the rational numbers. One may generalize this to "closed-form numbers", which may be defined in various ways. Most broadly, all numbers that can be defined explicitly or implicitly in terms of polynomials, exponentials, and logarithms are called "elementary numbers", and these include the algebraic numbers, plus some transcendental numbers. Most narrowly, one may consider numbers explicitly defined in terms of polynomials, exponentials, and logarithms – this does not include all algebraic numbers, but does include some simple transcendental numbers such as e or ln 2.

Algebraic integers

Algebraic numbers colored by leading coefficient (red signifies 1 for an algebraic integer)

Main article: Algebraic integer

An algebraic integer is an algebraic number that is a root of a polynomial with integer coefficients with leading coefficient 1 (a monic polynomial). Examples of algebraic integers are and Therefore, the algebraic integers constitute a proper superset of the integers, as the latter are the roots of monic polynomials xk for all . In this sense, algebraic integers are to algebraic numbers what integers are to rational numbers.

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 that are algebraic integers are the integers, and because the algebraic integers in any number field are in many ways analogous to the integers. If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as OK. These are the prototypical examples of Dedekind domains.

Special classes


  1. ^ Some of the following examples come from Hardy & Wright (1972, pp. 159–160, 178–179)
  2. ^ Garibaldi 2008.
  3. ^ Also, Liouville's theorem can be used to "produce as many examples of transcendental numbers as we please," cf. Hardy & Wright (1972, p. 161ff)
  4. ^ Hardy & Wright 1972, p. 160, 2008:205.
  5. ^ Niven 1956, Theorem 7.5..
  6. ^ Niven 1956, Corollary 7.3..
  7. ^ Niven 1956, p. 92.