Finite Field Isomorphism Suppose $E$ and $E'$ are two degree-$n$ extensions of $\mathbb{F}_p$. They are both splitting fields of $x^{p^n}-x$ and are isomorphic. Is it possible to obtain an isomorphism $E\to E'$ that fixes the base field $\mathbb{F}_p$?
 A: Any homomorphism between rings containing $\mathbb{F}_p$ fixes $\mathbb{F}_p$, because it must fix $1$, and every element of $\mathbb{F}_p$ is obtained by just adding $1$ to itself a bunch of times.
But more generally, the statement that any two splitting fields of a polynomial are isomorphic is a statement over the base field.  That is, more precisely, if $K$ is a field and $f\in K[x]$, and $E$ and $E'$ are two splitting fields of $f$ over $K$, then there is an isomorphism $E\to E'$ which fixes $K$ pointwise.  If you examine any proof of the statement "splitting fields of the same polynomial are isomorphic" which you may have seen, you will see that it actually proves this stronger statement.
A: Another way to look at this is, a finite field of order $\mathbb{F}_{p^{n}}$ has a subfield of order $p^{d}$ if and only if $d \mid n$, and if this is the case then the subfield of order $p^{d}$ is unique.  So (for $d=1$) there is only one subfield of $E$ isomorphic to $\mathbb{F}_{p}$, and the same for $E^{\prime}$.  Any isomorphism must send a subfield of order $p^{d}$ to a subfield of order $p^{d}$, so it fixed $\mathbb{F}_{p}$.
