Algebraic number

In mathematics, an algebraic number relative to a field (mathematics) F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form : a n x n + a n −1 x n −1 + ··· + a 1 x + a 0 = 0 where n is a positive integer called the degree of the polynomial, every coefficient a i is an element of F , and a n is nonzero. If the field F is the field Q of rational numbers and K is an algebraically closed field then the algebraic numbers relative to Q are simply called algebraic numbers. The algebraically closed field in which these numbers lie can be the complex numbers C, but sometimes other fields are used. Any such algebraic closure is unique up to field Isomorphism, but may differ in topological properties. Considered purely as a field it is unique, and it is either this abstract field devoid of topology or the closure of the rationals in the complex numbers which is most often called the field of algebraic numbers.

All rationals are algebraic. A real number number that is not rational may or may not be algebraic; for example irrational numbers such as 21/2 (the square root of 2) and 31/3/2 (the cube root of 3 divided by 2) are also algebraic because they are the solutions of x 2 − 2 = 0 and 8 x 3 − 3 = 0, respectively. But most real numbers are not algebraic; examples of this are Pi and e (mathematical constant) . If a complex number is not an algebraic number then it is called a transcendental number. So, for instance i , the imaginary unit, is an algebraic number since it satisfies x 2 + 1 = 0; however i^i is transcendental by the Gelfond-Schneider theorem; one branch of this number is e-π/2, which shows that eπ is also transcendental.

If an algebraic number satisfies 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 (mathematics), called the algebraic closure of the field of algebraic numbers. 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 field. 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 obtained from the integers using a . An example of such a number would be the unique real root of x 5 − x − 1 = 0.

=Algebraic integers=

An algebraic number which satisfies a polynomial equation of degree n with leading coefficient a n  = 1 (that is, a monic polynomial) and all other other coefficients a i belonging to the set Z of integers, is called an algebraic integer. Examples of algebraic integers are 3√2 + 5 and 6 i  - 2.

The sum, difference and product of algebraic integers are again algebraic integers, which means that the algebraic integers form a ring (algebra). 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 algebraic 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 O K.

=Special classes of algebraic number=

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