Let $F$ and $K$ be any two different fields and $p$ be a prime number. Let $X^q-X\in F[X]$ and $X^q-X\in K[X]$ where $q=p^k$ with integer $k>1$. Are the splitting fields of these polynomials necessarily isomorphic?
I think if $F$ is a finite field of order $q$ and $K$ is infinite field not containing all roots of the polynomial $X^q-X$ over $K$ then the splitting fields cannot be isomorphic. Because, in this case, $F$ is the splitting field of $X^q-X\in F[X]$ that is finite. And the splitting field of $X^q-X\in K[X]$ is infinite.
Can anyone check my arguement? Thanks!
Your counterexample is correct. If $F$ and $K$ really are any two fields at all, bearing no relationship to one another, then there's no reason why any particular extensions of them ought to be isomorphic.