How can I prove that you can not solve the polynomial : x^8-x^7+x^5-x^4+x^3-x+1
(No use by Eisenstein theorem , without using polynomial Cyaklotomi))
Hmm, are you the OP in this thread from the Math Is Fun forum?
According to the Rational Root Theorem:
Let be a polynomial equation with integer coefficients. If has a rational root , with relatively prime integers, then and . (Statement found in PlanetMath: rational root theorem)
In our case, and . Assume there is a rational root , then and . Now, , and so . (Two case after all!)
Checking the case we have and , contradicting .