noun 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+ib is an algebraic integer since it is a solution of either the equation z2+(−2a)z+(a2+b2)=0 or the equation z−a=0.