###CONJECTURE

Let $A= (c0,c1,..,cn)$ be a circulant matrix, i.e if $(c0,c1,..,cn)$ is the first column of $A$ then the $i$^th column of $A$ is obtained by applying the permutation $(1,2,..,n)^{n-1}$.

Assume $A$ in $GL_n(Z)$, i.e $A$ with integer entries and determinant +/-1 and moreover $c0+c1+..+cn=+-1$.

Then there exists one $j$ such that $cj=+/-1$ and $ci=0$ for all $i$ different from $j$.

Is this conjecture true?

What if we add the assumption that $n=p$ a prime?

Thanks for any idea! Fabienne