I have the following situation: let $m,n$ be integers such that $m|n$ and let $\zeta_m$, $\zeta_n$ denote primitive $m$ and $n$th roots of unity. Then we have the inclusion of fields
$\mathbb{Q}\subset \mathbb{Q}(\zeta_m) \subset \mathbb{Q}(\zeta_n)$ Now suppose we also have primes (where $(p,n)=1$) $(p)\subset \mathbb{Z}$ and then $\mathfrak{p}\subset \mathbb{Z}(\zeta_m)$ lying over $(p)$ and $\mathfrak{P}\subset \mathbb{Z}(\zeta_n)$ lying over $\mathfrak{p}$.
I have a congruence in $\mathbb{Q}(\zeta_n)$ of the form $a\equiv b \pmod{\mathfrak{P}}$, where $a,b$ are actually elements of $\mathbb{Q}(\zeta_m)$.
What can I say about the congruence properties of $a,b$ in $\mathbb{Q}(\zeta_m)$? Also, if I take the trace or the norm down to $\mathbb{Q}$, can I say anything about their congruence properties there?
Thanks!