Let $E$ and $F$ be fields, and define the map $*^{\sigma} : F[x] \rightarrow F'[x]$, which simply applies $\sigma$ to the coefficients of a polynomial $f(x)$. It is an isomorphism of rings.
Let $\sigma : F \rightarrow F'$ be a field isomorphism, and let $f \in F[x]$ be irreducible. Let $F(\alpha)$ be the field generated by $\alpha$ over $F$.
If $\alpha \in E/ F$ is a root of $f(x)$ and $\alpha' \in E'/F'$ is a root of $f^{\sigma}(x)$, then we can extend $\sigma$ to $\sigma' : F(\alpha) \rightarrow F'(\alpha')$, with $\sigma'(\alpha) = \alpha'$.
Now, while trying to prove this, I did the following;
I know that $\frac{F[x]}{<f>} \simeq F(\alpha)$ and $\frac{F'[x]}{<f^{\sigma}>} \simeq F(\alpha)$. Let $\mu$ be the map for the first isomorphism and let $\gamma$ be the map for the second isomorphism.
I can then define a function $\tau : \frac{F[x]}{<f>} \rightarrow \frac{F'[x]}{<f^{\sigma}>} $ by sending $g(x) + <f> \rightarrow g^{\sigma}(x) + <f^{\sigma}>$, which is an isomorphism.
So there is an isomorphic map from $F(\alpha)$ to $F'(\alpha')$ given by the composition of $\gamma(\tau(\mu^{-1}(\alpha))$.
In order to check that we've extended $\sigma$ to $\sigma'$ and $\sigma'(\alpha) = \alpha'$, we need to know what the maps $\mu$ and $ \gamma$ are. How are these maps defined?
Also, what does extending an isomorphism mean? How would one check that $\sigma$ has been 'extended' to $\sigma'$?
Thank you for any help.