Algebraic number

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An algebraic number is any real or complex number that is a solution of a polynomial equation of the form

anxn + an-1xn-1 + ... + a1x1 + a0 = 0

where n >= 0 and every ai is an integer.

All rational numbers are algebraic because every fraction a / b is a solution of b'x - a = 0. Some irrational numbers such as 21/2 (the square root of 2) and 31/3 (the cube root of 3) are also algebraic because they are the solutions of x2 - 2 = 0 and x3 - 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 with a polynomial of degree n and not such an equation with a lower degree than the number is said to be an algebraic number of degree n.

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 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.


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