Let $m$ be a natural number and $\zeta_m$ be a primitive $m$th root of unity.
Let $G_m$ be the galois group of $\mathbb{Q}(\zeta_m)$ over $\mathbb{Q}$.
Let $V_m$ be the multiplicative group generated by $$\{\pm \zeta_m , 1-\zeta_m ^a | 1 \leq a \leq m-1 \}.$$
And also we define $D_m = \{(1- \zeta_m) ^{r_m} | r_m \in \mathbb{Z} [G_m] \} $. Due to Bass, it is well-known that the $\mathbb{Z}$-rank of $V_m$ is $\frac 12 \phi (m) + \pi (m) -1 $, where $\pi (m)$ is the number of distinct primes dividing $m$.
Clearly we know that $V_m$ contains $D_m$. Also I already know that $V_m$ equals $D_m$ when m is a power of a prime. Now in my study, I have to compute the $\mathbb{Z}$-rank of $D_m$ and compare with $V_m$. But I am not sure how to compute it. If I get your advice on the approach, I sincerely appreciate.