What is the definition of the splitting field of a $p$-group? As I was reading an article I saw this statement :
Let $P$ be an extra special $p$-group and $\mathbb{F}$ be a field with char$(\mathbb{F})\neq p$ that is a splitting field for $P$.
At first I thought it is equivalent to say that $\mathbb{F}$ contains a primitive $|P|$th root of unity but then I realised that if $\mathbb{F}$ contains a primitive $|P|$th root of unity then char($\mathbb{F}$) cannot be equal to $p$ so the first assumption is unnecessary.