Let p be an odd prime integer and let $\zeta$ be a primitive pth root of unity. Let $\alpha$ be a nontrivial cyclotomic unit of $\mathbb Q(\zeta)$, i.e. an element of the form $1+\zeta+\dots+\zeta^{r1}$, with $2\le r < p$. Does $\alpha$ generate a normal basis of $\mathbb{Q}(\zeta)$? In other words does the Galois conjugates of $\alpha$ form a basis of $\mathbb Q(\zeta)$?

$\begingroup$ Clearly I asked the wrong question. I posted another question with the real question I am interested in. $\endgroup$ – Angel del Rio Apr 6 '15 at 15:35

$\begingroup$ The answer is positive. I got a solution by René Schoof $\endgroup$ – Angel del Rio Jul 29 '15 at 12:02
No, in general $\alpha$ doesnt generate a normal basis. The smallest counterexample is $p=5$, $r=3$. In fact for all $p\geq 5$, $r=(p+1)/2$ gives a counterexample: For $1\leq i\leq p1$, let $\varphi_i\in \text{Gal}(\mathbb{Q}(\zeta)/\mathbb{Q})$ be the automorphism given by $\zeta\mapsto \zeta^i$. A small computation shows that $\varphi_i(\alpha)+\varphi_{pi}(\alpha)=1$ for all $i$. In particular we have $\alpha\varphi_2(\alpha)\varphi_{p2}(\alpha)+\varphi_{p1}(\alpha)=0$ and hence the conjugates of $\alpha$ are not linearly independent for $p\geq 5$.
There are other counterexamples as well: For $p=11$ the counterexamples are given by $r=3$, $r=6$ and $r=9$.