Do two isomorphic finite field extensions have the same dimension?

Question

If $E = F(u_1, \cdots u_n) \cong \bar{E} = F(v_1, \cdots v_m)$ then do the two extensions necessarily have the same dimension over $F$?

Answer 1

Hint: Claim that if $E$ is a field extension over $F$ and $\sigma$ is a homomorphism from $E$ into $E'$, then $[E^{\sigma}:F^{\sigma}]=[E:F]$. You can prove this by showing that $\{v_1,v_2,\ldots ,v_n\}\text{ is a basis of E over F}\iff\{\sigma(v_1),\sigma(v_2),\ldots ,\sigma(v_n)\}\text{ is a basis of }$ $E^{\sigma}$ over $F^{\sigma} $

Now what happens in the case when $\sigma$ is an isomorphism of $E$ onto $E'$ and fixes $F$?