Algebraic number

An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients.

, is an algebraic number, because it is a root of the polynomial x2 − x − 1.

Real and complex numbers that are not algebraic, such as π and e, are called transcendental numbers.

The set of algebraic (complex) 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.

Similarly, the set of algebraic (real) numbers is countably infinite and has Lebesgue measure zero as a subset of the real numbers, and in that sense almost all real numbers are transcendental.

, is of finite degree if and only if α is an algebraic number.

, each can be expressed as sums of products of rational numbers and powers of α, and therefore this condition is equivalent to the requirement that for some finite

or, equivalently, α is a root of

; that is, an algebraic number with a minimal polynomial of degree not larger than

It can similarly be proven that for any finite set of algebraic numbers

The sum, difference, product, and quotient (if the denominator is nonzero) of two algebraic numbers is again algebraic: For any two algebraic numbers α, β, this follows directly from the fact that the simple extension

, is a linear subspace of the finite-degree field extension

An alternative way of showing this is constructively, by using the resultant.

Algebraic numbers thus form a field[7]

In fact, it is the smallest algebraically closed field containing the rationals and so it is called the algebraic closure of the rationals.

That the field of algebraic numbers is algebraically closed can be proven as follows: Let β be a root of a polynomial

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, ≈ 1.1673, that cannot be expressed in terms of only radicals and arithmetic operations.

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.

An algebraic integer is an algebraic number that is a root of a polynomial with integer coefficients with leading coefficient 1 (a monic polynomial).

Therefore, the algebraic integers constitute a proper superset of the integers, as the latter are the roots of monic polynomials x − k 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.

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.

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.
Algebraic numbers on the complex plane colored by degree (bright orange/red = 1, green = 2, blue = 3, yellow = 4). The larger points come from polynomials with smaller integer coefficients.
Algebraic numbers colored by degree (blue = 4, cyan = 3, red = 2, green = 1). The unit circle is black. [ further explanation needed ]
Visualisation of the (countable) field of algebraic numbers in the complex plane. Colours indicate the leading integer coefficient of the polynomial the number is a root of (red = 1 i.e. the algebraic integers, green = 2, blue = 3, yellow = 4...). Points becomes smaller as the other coefficients and number of terms in the polynomial become larger. View shows integers 0,1 and 2 at bottom right, +i near top.