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Algebraic number
In mathematics, an algebraic number relative to a field 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_{0}x^{n} + a_{1}x^{n−1} + ··· + a_{n −1}x + a_{n} = 0
where n is a positive integer called the degree of the polynomial, every coefficient a_{i} is an element of F, and a_{0} 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 which is not rational may or may not be algebraic; for example 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 x^{2} − 2 = 0 and 8x^{3} − 3 = 0, respectively. But most real numbers are not algebraic; examples of this are π and e. 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; this number is e^{-π/2}, which shows that e^{π} 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.
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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, 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. 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. 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 as above with a_{0} = 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, 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.
Special classes of algebraic number
- Gaussian integer
- Eisenstein integer
- Quadratic irrational
- Fundamental unit
- Root of unity
- Gaussian period
- Pisot-Vijayaraghavan number
- Salem number