No, in general $\alpha$ doesnt generate a normal basis. The smallest counterexample is $p=5$, $r=3$. In fact my computations indicate that for all $p\geq 5$, $r=(p+1)/2$ always gives a counterexample. But there are other counterexamples as well: For $p=11$ the counterexamples are given by $r=3$, $r=6$ and $r=9$.
Kasper Andersen
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