Algebraic number

In mathematics, an algebraic number is any real or complex number that is a solution of a polynomial equation of the form

a_nxⁿ + a_n−1xⁿ⁻¹ + ··· + a₁x + a₀ = 0

where n > 0, every a_i is an integer, and a_n is nonzero.

All rational numbers are algebraic because every fraction a / b is a solution of bx − a = 0. Some irrational numbers such as 2^1/2 (the square root of 2) and 3^1/3/2 (the cube root of 3 divided by 2) are also algebraic because they are the solutions of x² − 2 = 0 and 8x³ − 3 = 0, respectively. But not all real numbers are algebraic. Examples of this are π and e. If a complex number is not an algebraic number then it is called a transcendental number.

If an algebraic number satisifies such an equation as given above with a polynomial of degree n and not such an equation with a lower degree, then the number is said to be an algebraic number of degree n.

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. It can be shown that if we allow the coefficients a_i to be any algebraic numbers then every solution of the equation will again be an algebraic number. This 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 is therefore called the algebraic closure of the rationals.

Numbers defined by radicals

All numbers which can be written using a finite number of additions, subtractions, multiplications, divisions, and n^th roots (where n is a positive integer) are algebraic. The converse, however, is not true: there are algebraic numbers which cannot be written in this manner. All of these numbers are solutions to polynomials of degree ≥ 5. This is a result of Galois theory.

Algebraic integers

An algebraic number which satisfies a polynomial equation of degree n as above with a_n = 1 (that is, a monic polynomial), is called an algebraic integer. Examples of algebraic integers are 3√2 + 5 and 6i - 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.

If K is a number field, its ring of integers is the subring of algebraic integers in K.

Special classes of algebraic number

• Gaussian integer
• Eisenstein integer
• Quadratic irrational
• Fundamental unit
• Root of unity
• Gaussian period
• Pisot-Vijayaraghavan number
• Salem number

More general situations

Both the notions of algebraic number and algebraic integer may be usefully generalized to fields other than the complex numbers; see algebraic extension and integral closure.