In mathematics, an algebraic number is a complex number that is a root of a non-zero polynomial in one variable with rational (or equivalently, integer) coefficients. Numbers such as π that are not algebraic are said to be transcendental, and are infinitely more numerous within the complex number field.
The sum, difference, product and quotient of two algebraic numbers is again algebraic (this non-obvious fact can be demonstrated using the result...
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In mathematics, an algebraic number is a complex number that is a root of a non-zero polynomial in one variable with rational (or equivalently, integer) coefficients. Numbers such as π that are not algebraic are said to be transcendental, and are infinitely more numerous within the complex number field.
The sum, difference, product and quotient of two algebraic numbers is again algebraic (this non-obvious fact can be demonstrated using the resultant), and the algebraic numbers therefore form a field, sometimes denoted by (which may also denote the adele ring) or . 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.
All numbers which can be obtained from the integers using a finite number of additions,...
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