algebraic integer

English edit

Noun edit

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 $z=a+ib$ is an algebraic integer since it is a solution of either the equation $z^{2}+(-2a)z+(a^{2}+b^{2})=0$ or the equation $z-a=0$ .
- 1984, Alan Baker, A Concise Introduction to the Theory of Numbers, Cambridge University Press, page 62:
  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 $R$ .
- 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 $O_{d}$ be the set of algebraic integers in an imaginary quadratic number field $\mathbb {Q} [{\sqrt {d}}],\ d<0$ , where $d$ is the discriminant of $O_{d}$ .
- 2010, Pierre Moussa, “Localisation of algebraic integers and polynomial iteration”, in Sergiy Kolyada, Yuri Nanin, Martin Möller, Pieter Moree, Thomas Ward, editors, Dynamical Numbers: Interplay Between Dynamical Systems and Number Theory, American Mathematical Society, page 83:
  We consider the problem of finding all algebraic integers which belong to a bounded subset of the complex plane together with their conjugates.