Algebraic number

From Example Problems
Jump to navigation Jump to search

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

anxn + an−1xn−1 + ··· + a1x + a0 = 0

where n is a positive integer called the degree of the polynomial, every coefficient ai is an element of F, and an 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 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 x2 − 2 = 0 and 8x3 − 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 x2 + 1 = 0; however Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "":): {\displaystyle 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, called the algebraic closure of the field of algebraic numbers. It can be shown that if we allow the coefficients ai 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 obtained from the integers using a finite number of additions, subtractions, multiplications, divisions, and nth 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 x5 − x − 1 = 0.

Algebraic integers

An algebraic number which satisfies a polynomial equation of degree n with leading coefficient an = 1 (that is, a monic polynomial) and all other other coefficients ai belonging to the set Z of integers, 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, and is frequently denoted as OK.

Special classes of algebraic number

cs:Algebraické číslo da:Algebraiske tal de:Algebraische Zahl es:Número algebraico fr:Nombre algébrique gl:Número alxebraico ko:대수적 수 it:Numero algebrico he:מספר אלגברי nl:Algebraïsch getal ja:代数的数 pl:Liczby algebraiczne pt:Número algébrico ru:Алгебраическое число zh:代數數