algebraic number

== English == === Noun === algebraic number (plural algebraic numbers) (algebra, number theory) A 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 x 2 − x − 1 = 0 {\displaystyle x^{2}-x-1=0} , whose coefficients are integers. The square root of a rational number, m n {\displaystyle \textstyle {\sqrt {\frac {m}{n}}}} , is an algebraic number since it is a solution of the quadratic equation n x 2 − m = 0 {\displaystyle nx^{2}-m=0} , whose coefficients are integers. 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: (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). ==== Hyponyms ==== algebraic integer phi, golden ratio ==== Coordinate terms ==== transcendental number ==== Derived terms ==== ==== Translations ==== === See also === algebraic integer === Further reading === Algebraic number field on Wikipedia.Wikipedia Algebraic integer on Wikipedia.Wikipedia Algebraic number on Encyclopedia of Mathematics algebraic number on nLab Algebraic Number on Wolfram MathWorld