Proving that a Galois group $Gal(E/Q)$ is isomorphic to $\mathbb{F}_p^\times$
If, as you say, you already know what you wrote there, then you've already proved what you need...almost.
Let $\;\zeta=e^{2\pi i/p}\;$ , then $\;\zeta\;$ is clearly a root of $\;f(x)=x^p-1\;$ . Observe (well, prove) now that
1) All the roots of $\;f(x)\;$ are different (hint: look at $\;f'(x)\;$ )
2) $\;\forall\,k\in\Bbb Z\;,\;\;\;\left(\zeta^k\right)^p=1\;$
Deduce then that $\;[\Bbb Q(\zeta):\Bbb Q]=p-1\;$ and that $\;\Bbb Q(\zeta)\;$ is the splitting field of $\;f(x)\;$ over the rationals. Also, from (2) it follows this extension is cyclic.