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- (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 is an algebraic integer since it is a solution of either the equation or the equation .
- 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 in the minimal polynomial is 1. The algebraic integers in an algebraic number field form a ring .
- 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 be the set of algebraic integers in an imaginary quadratic number field , where is the discriminant of .
- 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.
- cyclotomic integer
- phi, golden ratio
- quadratic integer
- root of unity
algebraic number whose minimal polynomial has integer coefficients