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algebraic integer

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algebraic integer

出典:『Wiktionary』 (2025/10/23 21:00 UTC 版)

名詞

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 z = a + i b {\displaystyle z=a+ib} {\displaystyle z=a+ib} is an algebraic integer since it is a solution of either the equation z 2 + ( − 2 a ) z + ( a 2 + b 2 ) = 0 {\displaystyle z^{2}+(-2a)z+(a^{2}+b^{2})=0} {\displaystyle z^{2}+(-2a)z+(a^{2}+b^{2})=0} or the equation z − a = 0 {\displaystyle z-a=0} {\displaystyle z-a=0}.
    • 1984, Alan Baker, A Concise Introduction to the Theory of Numbers, Cambridge University Press, page 62:
    • 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 {\displaystyle O_{d}} {\displaystyle O_{d}} be the set of algebraic integers in an imaginary quadratic number field Q [ d ] , d < 0 {\displaystyle \mathbb {Q} [{\sqrt {d}}],\ d<0} {\displaystyle \mathbb {Q} [{\sqrt {d}}],\ d<0}, where d {\displaystyle d} {\displaystyle d} is the discriminant of O d {\displaystyle O_{d}} {\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:

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