Let be a field and let be a nonzero algebraic element over . Prove that the conjugates of are nonzero.
Let be the minimal polynomial of over . Recall that the conjugates of are precisely the roots of . If is linear, then the only conjugate of is itself. Suppose has degree at least 2. If one of the roots of is 0, then in fact ; note that is a root of and that has degree strictly smaller than that of , violating the minimalness of . So 0 is not a conjugate of .