How do I prove the following fact about splitting fields?

Question

Hello I have the following problem:

Let $F$ be a splitting field of the polynomial $g(X)\in k[X]$. If $E$ is another splitting field of $g$ then there exists an isomorphism $\psi:E\rightarrow F$ inducing the identity on $k$.

I somehow do not see why this should be true. So I mean I know that $g$ splits in $E,F$ and that $E=k(\alpha_1,...,\alpha_n)$, $F=k(\beta_1,...\beta_m)$ where $\alpha_i, \beta_j$ are the roots of $g$ in $E$ respectivly $F$ right? But I don't know if $m=n$ do I? because if $m=n$ then I would define $\psi(\alpha_i)=\beta_i$ which is clearly an isomorphism but I'm not sure if this can be done.

Could someone help me?

Thanks for your help.

Answer 1

"But I don't know if m=n do I?"

This is a degree argument. We may assume that $g(x)$ is monic. Then $g(x) = \prod_{i=1}^n (x-\alpha_i) = \prod_{j=1}^m (x-\beta_j)$ and so (by the definition when two polynomials are equal) $m=n$.