Galois Group does not depend on the choice of the field extension

Question

Let $K$ be a field, and a polynomial $f\in K[X]$. Let $E$ and $F$ be two splitting fields of $f$ (albeit isomorphic), to show is two Galois groups Gal$(E/K)$ and Gal$(F/K)$ are isomorphic.

One idea could be the Galois groups also permute the roots. We can prove that it is possible to construct the isomorphism between two splitting fields match some roots in pair. Use this this to induce the isomorphism between two Galois groups.

Is there any quick proof?

Answer 1

Suppose that $\phi : E \to F$ is a field isomorphism between the two splitting fields. Then $$\begin{array}{l|rcl} \Phi : & \text{Gal}(E/K) &\longrightarrow & \text{Gal}(F/K) \\ & \sigma & \longmapsto & \phi \circ \sigma \circ \phi^{-1} \end{array}$$

is a group isomorphism between the Galois groups.