Algebraic number

In mathematics, an algebraic number is a real or complex number, the zero of a polynomial of degree x is greater than zero

With rational coefficients, ie, solution of the equation is.

The so-defined algebraic numbers form a proper subset of the complex numbers. Obviously any rational number is algebraically because it solves the equation. It is therefore necessary.

Is not a real algebraic (or more generally complex ) number, it is called transcendent.

The also common definition of algebraic numbers as zeros of polynomials with integer coefficients is given to the above equivalent. Each polynomial with rational coefficients can be converted by multiplying by the denominator of the coefficients in one with integer coefficients. The resulting polynomial has exactly the same zeros as the Ausgangspolynom.

Polynomials with rational coefficients can be normalized by dividing all coefficients by the coefficient. Zeros of the normalized polynomials whose coefficients are integers, called ganzalgebraische numbers or algebraic integers. For general concept of wholeness see wholeness ( commutative algebra).

One can extend the notion of algebraic number to that of the algebraic element by one, removes the coefficients of the polynomial instead of from any body.

Degree and minimal polynomial of an algebraic number

For many studies of algebraic numbers defined in the following degree and the minimal polynomial of an algebraic number are important.

If x is an algebraic number, which is an algebraic equation

With, met, but no such equation of a lower degree, then n is called the degree of x. This means that all rational numbers of degree 1 All irrational square roots of degree 2

The number n is the same, the degree of the polynomial f, the so-called minimal polynomial of x.

Examples

For example, an algebraic number, because it is a solution of the equation. Similarly, the imaginary unit as a solution of algebraically.

Towards the end of the 19th century proved that the circuit number and the Euler number are not algebraic. From other numbers, such as, you do not know to this day whether they are algebraic or transcendental. See article transcendental number.

Properties

The set of algebraic numbers is countable and forms a body.

The field of algebraic numbers is algebraically closed, ie, every polynomial with algebraic coefficients has only algebraic zeros. This body is a minimal algebraically closed upper body and is therefore an algebraic closure of. One often writes it as ( for " algebraic closure of Q"; confused with another statutory terms ) or a ( for " Algebraic Numbers").

Above the field of rational numbers and below the body of algebraic numbers are an infinite number of intermediate body; as the set of all numbers of the form, and wherein said rational numbers, and the square root of a rational number. Also the body of ruler and compass constructible from points of the complex plane is one such algebraic intermediate body. → See Euclidean body.

In the context of Galois this intermediate body to be examined, so as to obtain a deep insight on the solubility or non- solubility of equations. One result of Galois theory is that, although each complex number, obtained from rational numbers by using the basic arithmetic operations (addition, subtraction, multiplication and division ) as well as by drawing n-th roots ( n is a natural number ) can be obtained ( we call such numbers ), is " representable by radicals" algebraically, but conversely there are algebraic numbers, which can not be represented in this way; all these figures are zeros of polynomials of degree ≥ 5

Algebraic numbers that are even zeros of a polynomial with integer coefficients normalized, hot algebraic integers or ganzalgebraische numbers. The algebraic integers that are rational are exactly the integers. The algebraic integers form a subring of algebraic numbers. For the wholeness in general, see wholeness ( commutative algebra).

• Number
• Special number
• Algebraic Number Theory
• Body theory