Is it possible for two distinct irreducible polynomials with integer coefficients to have a root in common? In other words, is it possible that a root is shared by some two distinct irreducible polynomials?
I would say no, because then it could be divided out by GCD of the polynomials, which would contradict their irreducibility but maybe I'm wrong?
Conversely, if two polynomials are irreducible (with positive leading coefficient) and share a root, they must be the same (i.e. they must share all the roots)?