I am reading a paper which states without proof that, for $r \ge 2$, the polynomial $f(X) = X^r – X^{r-1}- X^{r-2} – \ldots – 1$ has no roots (in $\Bbb{C}$) that are roots of unity. After playing around with elementary algebra and getting nowhere, I tried to apply the methods of this note by Keith Conrad on polynomials with roots on the unit circle , but with no success. I suspect there is something very simple that I am missing. Any clues about how to prove this will be very gratefully received.
Polynomial with no roots of unity among its roots
polynomialsrootsroots-of-unity
Best Answer
Notice that $g(x)=(x-1)f(x)=x^{r+1}-2x^{r}+1$ and look at its zeroes on the unit circle (we know $\alpha=1$ is a root of $g$ but clearly not of $f$, we will show there is no other). So let $g(\alpha)=0$ and $|\alpha|=1$, then $$ 1=|\alpha^{r+1}-2\alpha^r|=|\alpha|^r|\alpha-2|=|\alpha-2|. $$ Only complex number that satisfies both $|\alpha|=1$ and $|\alpha-2|=1$ is $\alpha=1$ (can be seen either as an intersection of two circles in the complex plane, or solved fully algebraically if needed). So $g(x)$ has no other roots on the unit circle than the $\alpha=1$, and by the construction and the fact that $f(1) \neq 0$, $f(x)$ has no roots on the unit circle.