algebraic integer

English

Noun

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+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.
    • 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 Od be the set of algebraic integers in an imaginary quadratic number field ℚ[d], d<0, where d is the discriminant of Od.
    • 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.

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Holonyms

  • ring of integers

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Further reading

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