I have to proof that for all $p$ with $p$ prime the following holds: $p| 1 + (p-1)!$. I have to use the following fact: $\Pi_{x \in \mathbb{F}_{q}^{*}} x = -1$.
This is my proof:
We know that $\mathbb{F}_{p}$ is a finite field. So $\Pi_{x \in \mathbb{F}_{p}^{*}} x = -1$. We also notice that $(p-1)! = \Pi_{x\in\mathbb{Z}_{p}^{*}}x.$ This implies dat $(p-1)! = -1$ mod $p$ $\implies (p-1)! + 1 = 0$ mod $p$. The last statement proofs that $p| 1 + (p-1)!$.
Is this reasoning correct? I'm also still confused If I can jump from $\mathbb{F}_{p}^{*}$ to $\mathbb{Z}_{p}^{*}$ and with $\mathbb{Z}_{p}^{*}$ I mean the integers modulo $p$. I just started to learn about finite fields, so every comment is more than welcome.
