# algebraic integer

## English

### Noun

algebraic integer (plural algebraic integers)

1. () 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 ${\displaystyle z=a+ib}$ is an algebraic integer since it is a solution of either the equation ${\displaystyle z^{2}+(-2a)z+(a^{2}+b^{2})=0}$ or the equation ${\displaystyle z-a=0}$.
• 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 ${\displaystyle x}$ in the minimal polynomial ${\displaystyle P}$ is 1. The algebraic integers in an algebraic number field ${\displaystyle k}$ form a ring ${\displaystyle R}$.
• 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 ${\displaystyle O_{d}}$ be the set of algebraic integers in an imaginary quadratic number field ${\displaystyle \mathbb {Q} [{\sqrt {d}}],\ d<0}$, where ${\displaystyle d}$ is the discriminant of ${\displaystyle O_{d}}$.
• 2010, Pierre Moussa, Localisation of algebraic integers and polynomial iteration, 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.