I'm looking at chapter 4 of Waterhouse's "Abelian varieties over finite fields"; and Theorems 4.1 and 4.2 seem to use the following fact:

Suppose that $E/\mathbb{F}_q$ is an elliptic curve over a finite field with $q=p^n$ elements and let $\pi_E$ denote the $q$th-power Frobenius map acting on $E$. Suppose that $\pi_E$ does not act like multiplication-by-$N$ for any integer $N$ so that $\pi_E = [\alpha]_E$ where $\alpha$ is a root of the polynomial $$x^2 - \mathrm{tr}(\pi_E)x + q$$ Then $E$ is ordinary if and only if $p$ is splits in $\mathbb{Q}(\alpha)$.

Waterhouse uses some very general theory (namely a theorem of Honda/Tate and the general theory of abelian varieties) and I find his reasoning/terminology vague. Does anybody know of a more down to earth proof for this result?

I've tried using the fact that $E$ is ordinary if and only if $q\nmid \mathrm{tr}(\pi_E)^2$.

If $p$ splits in $\mathbb{Q}(\alpha)$, then $(p) = \mathfrak{p}\overline{\mathfrak{p}}$ and thus (as Waterhouse points out) $(\alpha) = \mathfrak{p}^a\overline{\mathfrak{p}}^b$ where $a+b=n$. Not sure what to do from here.