algebraic number

plural algebraic numbers

1. algebraA complex number (more generally, an element of a number field) that is a root of a polynomial whose coefficients are integers; equivalently, a complex number (or element of a number field) that is a root of a monic polynomial whose coefficients are rational numbers.

   The golden ratio (φ) is an algebraic number since it is a solution of the quadratic equation x²-x-1=0, whose coefficients are integers.

   4 more examples

   The square root of a rational number, √(m/n), is an algebraic number since it is a solution of the quadratic equation nx²-m=0, whose coefficients are integers.

   Thus, the equation x-eʸ#61;0 is satisfied for x#61;1,#92;y#61;0 and for no other pair of algebraic numbers.

   1918, The American Mathematical Monthly, volume 25, Mathematical Association of America, page 435

   As a matter of fact, it is not true that the algebraic numbers above can be factored uniquely, but the first case of failure occurs when p = 23.

   1921, L. J. Mordell, Three Lectures on Fermat's Last Theorem, page 16

   (i) It is easily checked that the set of all algebraic numbers is countable, whereas the set of all complex numbers is uncountable (this non-constructive proof goes back to Cantor).

   1991, P. M. Cohn, Algebraic Numbers and Algebraic Functions, Chapman & Hall, page 83, The existence of such 'transcendental' numbers is well known and it can be proved at three levels