algebraic integer

Definitions

1. 1

   A real or complex number (more generally, an element of a number field) which is a root…

   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.
   • An algebraic number is said to be an algebraic integer if the coefficient of the highest power of x in the minimal polynomial P is 1. The algebraic integers in an algebraic number field k form a ring R.

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