Do you know we are in the same class in set theory? Unbelievable, I know.

Because needs to a zero of theminimal polynomialof , i.e. either . While it is true that both solve it need not to be true because if we restrict out consideration to minimal polynomials.

I want to add, the theorem: "automorphisms permute zeros of a polynomial" is necessary but not sufficient. Meaning if you have a zero then it must be mapped into its permuted zero. The other way around: that if you map a zero into a permuted zero, does not mean that you have an automorphism.