algebraic integer

algebraic integer (plural algebraic integers)

1. (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 ${\displaystyle z=a+ib}$ 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”, 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.

• algebraic number

• cyclotomic integer
• phi, golden ratio
• quadratic integer
  • Gaussian integer, Eisenstein integer
    • integer, rational integer
• root of unity

• ring of integers

• quadratic integer

• 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