Finite Field Isomorphism

Question

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 $\mathbb{F}_p$?

Answer 1

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.

Answer 2

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 $\mathbb{F}_{p}$.