I am self-studying Galois theory and I am now reading splitting fields.I am having difficulty in understanding the proof of the fact that any two splitting fields of a polynomial $f(X)\in F[X]$ are $F$-isomorphic.Some idea that I got by reading various texts is described in the following:
Suppose $L_1$ and $L_2$ be two splitting fields of the polynomial $f(X)$.For simplicity suppose $f(X)$ is a degree $2$ polynomial and we have two splitting fields $L_1=F(a_1,a_2)$ and $L_2$.Now we have the identity map $\iota:F\to F$.We want to extend this map to an $F$-isomorphism $\sigma_1:F(a_1\to F(b_1)$,where $b_1$ is a root of that irreducible factor $h_2(X)$ of $\iota(f(X))=f(X)$ that is the image of the irreducible factor $h_1(X)$ of $f(X)$ under $\iota$.Then we will extend this map to an $F$-isomorphism $\sigma:F(a_1,a_2)\to F(b_1,b_2)$.I think I understand how we should extend the map $\iota$.Suppose, $L_1$ is the splitting field of $f(X)$ over $F(a_1)$ and $L_2$ is the splitting field of $\iota(f(X))$ over $F(b_1)$ and we got an $F$-isomorphism from $F(a_1)$ to $F(b_1)$,say $\varphi$.Now,we will take $a_2$ and consider its minimal polynomial $m(X)$ over $F(a_1)$.We will quotient $F[X]$ by $(m(X))$.This gives rise to an isomorphism from $F(a_1,a_2)$ to $F(b_1)[X]/(\varphi(m(X))\simeq_F F(b_1,b_2)$.Is my idea correct?