/sci/ - Science & Math

Lads.. I don't get it. Why is [math]\text{Gal}(K/F)[/math] the identity. Can we not have some automorphism that permutes the extra roots? I would have thought if anything, it would be [math]S_n[/math]. Note that the minimal polynomial of [math]a_i\in F[/math] is denoted [math]\text{min}(F,a_i)[/math].

Can an automorphism fixing the ground field of an algebraic extension only permute the elements that belong to the same minimal polynomial?