Let $F=GF(p)$ be the field with $p$ elements.

Let $G$ be a finite group with order divisible by $p$.

Let $S$ be a fixed Sylow $p$-subgroup of $G$ and let [$P_i$] be a list of representatives of $p$-subgroups of $S$ up to conjugacy in $G$ (including $S$ and the trivial group).

For all $i$, let $N_i$ be the normalizer of $P_i$.

Is there an easy way to find the smallest (or a very small) extension field of $F$ with the property that it is a splitting field for all quotient groups $N_i/P_i$ (including $G\cong G/ \langle 1 \rangle$)?

One could take the splitting field of the polynomial $f(x):= x^m - 1 \in F[x]$, where $m$ is the exponent of the group $G$, but are there better choices (in general)?


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