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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, and an is nonzero.

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 31/3/2 (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 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 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 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.

All numbers which can be written starting from the rationals using only the arithmetical operations +,-,*,/ and square roots, cube roots etc. are algebraic. The converse however is not true: there are algebraic numbers which cannot be written in this manner. All of these numbers have degree ≥ 5. This is a result of Galois theory.

An algebraic number which satisfies a polynomial equation of degree n as above with an = 1 is called an algebraic integer. Examples of algebraic integers are 3×21/2 + 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.

Both the notions of algebraic number and algebraic integer may be usefully generalized to fields other than the complex numbers; see algebraic extension and integral closure.



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