Let $E/F$ be a finite Galois extension and $\alpha\in E$ and $p(t)=irr(\alpha,F)$, the monic irreducible polynomial defined over $F$ which has a root $\alpha$.
Let $\beta\in E$ be another root of $p(t)$.
My question arises here.
Are the two Galois groups $Gal(E/F(\alpha))$ and $Gal(E/F(\beta))$ isomorphic?
If this is not the case, could you suggest me some counter-example?
If they are always isomorphic, how can we detect the differences between each roots of an irreducible polynomial? Because, I know the all the roots of some irreducible polynomial are not of equal status from the view of total field $E$, not from the base field $F$.