Because needs to a zero of the minimal polynomial of , 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.