🚀 go-pugleaf

Let $p$ be an odd prime, and assume $\beta$ is an algebraic
integer of the form $$\beta = \sum_\omega \mu_\omega (1-\omega),$$
where the coefficients $\mu_\omega$ are nonnegative rational
integers and $\omega$ ranges over the primitive complex $p$-th
roots of unity.  Assume further that $\beta$ is irrational,
so the coefficients don't assume a constant value, that the
coefficients of $\beta$ are relatively prime, and that
the product of the algebraic conjugates of $\beta$ is a power
of $p$ (so the norm of $\beta$, in the field-theoretic sense,
is a power of $p$).  Does it necessarily follow that $\beta$
has the form $1-\omega$ or $2 Re(1-\omega)$?

Any information on this problem would be helpful. 
Please respond via e-mail to alvisd@natasha.iusb.indiana.edu