exclusion of integer root
Theorem. The equation
with integer coefficients has no integer roots (http://planetmath.org/Equation), if and are odd.
Proof. Make the antithesis, that there is an integer such that . This cannot be even, because else all terms of except were even and thus the whole sum could not have the even value 0. Consequently, and also its powers (http://planetmath.org/GeneralAssociativity) have to be odd. Since
there must be among the coefficients an odd amount of odd numbers. This means that
This however contradicts the assumption on the parity of , whence the antithesis is wrong and the theorem .