algebraic integer

== English == === Noun === algebraic integer (plural algebraic integers) (algebra, number theory) A real or complex number (more generally, an element of a number field) which is a root of a monic polynomial whose coefficients are integers; equivalently, an algebraic number whose minimal polynomial (lowest-degree polynomial of which it is a root and whose leading coefficient is 1) has integer coefficients. A Gaussian integer z = a + i b {\displaystyle z=a+ib} is an algebraic integer since it is a solution of either the equation z 2 + ( − 2 a ) z + ( a 2 + b 2 ) = 0 {\displaystyle z^{2}+(-2a)z+(a^{2}+b^{2})=0} or the equation z − a = 0 {\displaystyle z-a=0} . 1989, Heinrich Rolletschek, Shortest Division Chains in Imaginary Quadratic Number Fields, Patrizia Gianni (editor), Symbolic and Algebraic Computation: International Symposium, Springer, LNCS 358, page 231, Let O d {\displaystyle O_{d}} be the set of algebraic integers in an imaginary quadratic number field Q [ d ] , d < 0 {\displaystyle \mathbb {Q} [{\sqrt {d}}],\ d<0} , where d {\displaystyle d} is the discriminant of O d {\displaystyle O_{d}} . ==== Hypernyms ==== algebraic number ==== Hyponyms ==== cyclotomic integer phi, golden ratio quadratic integer Gaussian integer, Eisenstein integer integer, rational integer root of unity ==== Holonyms ==== ring of integers ==== Related terms ==== quadratic integer ==== Translations ==== === Further reading === algebraic integer on Wikipedia.Wikipedia Algebraic number on Wikipedia.Wikipedia Eisenstein integer on Wikipedia.Wikipedia Gaussian integer on Wikipedia.Wikipedia Integral element on Wikipedia.Wikipedia algebraic integer on nLab Algebraic Integer on Wolfram MathWorld