# How to find the smallest extension field of $GF(p)$ which is a splitting field for all quotient groups $N_i/P_i$?

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)?