multiples of an algebraic number

Theorem.  If α is an algebraic numberMathworldPlanetmath, then there exists a non-zero multipleMathworldPlanetmath ( of α which is an algebraic integerMathworldPlanetmath.

Proof.  Let α be a of the equation


where r1, r2, …, rn are rational numbersPlanetmathPlanetmathPlanetmath (n>0).  Let l be the least common multiple of the denominators of the rj’s.  Then we have


i.e. the the algebraic equation


with rational integer coefficients.

According to the theorem, any algebraic number ξ is a quotient ( of an algebraic integer (of the field (ξ)) and a rational integer.

Title multiples of an algebraic number
Canonical name MultiplesOfAnAlgebraicNumber
Date of creation 2014-05-16 19:58:36
Last modified on 2014-05-16 19:58:36
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Theorem
Classification msc 11R04
Related topic TheoryOfAlgebraicNumbers
Related topic AlgebraicSinesAndCosines
Related topic SomethingRelatedToAlgebraicInteger
Related topic RationalAlgebraicIntegers