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All rationals are algebraic. An Irrational Number may or may not be algebraic. For example, 21/2 (the Square Root Of 2 ) and 31/3/2 (half the Cube Root of 3) are algebraic because they are the solutions of ''x''2 − 2 = 0 and 8''x''3 − 3 = 0, respectively. The Imaginary Unit ''i'' is algebraic, since it satisfies ''x''2 + 1 = 0. The complex numbers that are not algebraic are called Transcendental Number s. Most complex numbers are transcendental, because the set of algebraic numbers is Countable while the set of transcendental numbers is not. Examples of transcendental numbers include π and '' E ''. Other examples are provided by the Gelfond-Schneider Theorem . All algebraic numbers are Computable and therefore Definable . If an algebraic number satisfies a polynomial 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 concept of algebraic numbers can be generalized to arbitrary Field Extension s; elements in such extensions that satify polynomial equations are called Algebraic Element s. 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 . It can be shown that every root of a polynomial equation whose coefficients are ''algebraic numbers'' is again algebraic. 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. It is sometimes denoted by . NUMBERS DEFINED BY RADICALS All numbers which can be obtained from the integers using a (see Quintic Equation s and the Abel–Ruffini Theorem ). An example of such a number would be the unique real root of ''x''5 − x − 1 = 0. See Also: algebraic integer An algebraic number which satisfies a Polynomial Equation of degree ''n'' with leading coefficient ''a''''n'' = 1 (that is, a Monic Polynomial ) and all other coefficients ''a''''i'' belonging to the set Z of Integer s, is called an ''' Algebraic Integer '''. Examples of algebraic integers are 3√ + 5 and 6''i'' - 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 ''O''K. These are the prototypical examples of Dedekind Domain s. SPECIAL CLASSES OF ALGEBRAIC NUMBER
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