Carmichael numbers and primitive roots of unity

Question

Let $n$ be a Carmichael number. Is it possible for an integer ring $\mathbb{Z}_n$ to contain primitive $(n-1)^{th}$ roots of unity? Or do only only primitive roots of unity of degree $\quad k < n-1 \quad$ exist in this case?

Answer 1

There are no primitive $(n-1)$th roots of unity in this ring. If there were, then it would follow that there are at least $n-1$ units modulo $n$, meaning that $n$ is prime.

In fact, if $n = p_1 p_2 \cdots p_m$, a product of $m$ primes (which must be distinct and odd, by a well-known fact about Carmichael numbers), then $\mathbb Z_n$ is isomorphic to $\mathbb Z_{p_1} \times \mathbb Z_{p_2} \times \cdots \times \mathbb Z_{p_n}$. Then the units of $\mathbb Z_n$ correspond to the ordered tuples of units in each $\mathbb Z_{p_i}$; i.e. the group of units of $\mathbb Z_n$ is the direct product of the groups of units of each $\mathbb Z_{p_i}$. This can't be a cyclic group, because it has more than one element of order two. In fact, there is a primitive $k$th root of unity if and only if $k$ divides the least common multiple of $p_1-1, p_2-1, \dots, p_m-1$.