exclusion of integer root

Theorem.  The equation

p(x):=anxn+an-1xn-1++a0= 0

with integer coefficients ai has no integer roots (http://planetmath.org/Equation), if p(0) and p(1) are odd.

Proof.  Make the antithesis, that there is an integer x0 such that  p(x0)=0.  This x0 cannot be even, because else all terms of p(x0) except a0 were even and thus the whole sum could not have the even value 0.  Consequently, x0 and also its powers (http://planetmath.org/GeneralAssociativity) have to be odd.  Since


there must be among the coefficients an,an-1,,a1 an odd amount of odd numbersMathworldPlanetmathPlanetmath.  This means that


This however contradicts the assumptionPlanetmathPlanetmath on the parity of p(1), whence the antithesis is wrong and the theorem .

Title exclusion of integer root
Canonical name ExclusionOfIntegerRoot
Date of creation 2013-03-22 19:08:21
Last modified on 2013-03-22 19:08:21
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 5
Author pahio (2872)
Entry type Theorem
Classification msc 12D10
Classification msc 12D05
Related topic DivisibilityInRings