rational algebraic integers

Proof.1.  Any rational integer m has the minimal polynomial x-m, whence it is an algebraic integer.
2.  Let the rational number  α=mn  be an algebraic integer where m,n are coprime integers and  n>0.  Then there is a polynomialPlanetmathPlanetmath


with  a1,,ak  such that

f(α)=(mn)k+a1(mn)k-1++ak= 0.

Multiplying this equation termwise by nk implies


which says that  nmk (see divisibility in rings).  Since m and n are coprimeMathworldPlanetmath and n positive, it follows that  n=1.  Therefore,  α=m.

Title rational algebraic integers
Canonical name RationalAlgebraicIntegers
Date of creation 2013-03-22 19:07:34
Last modified on 2013-03-22 19:07:34
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 5
Author pahio (2872)
Entry type Theorem
Classification msc 11R04
Related topic MultiplesOfAnAlgebraicNumber