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 
$$\mathfrak{p}\subset \mathbb{Q}(\zeta\_m)$$
 lying over $(p)$ and 
$$\mathfrak{P}\subset \mathbb{Q}(\zeta\_n)$$
 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 $\mathbb{Q}(\zeta\_m)$? More importantly, if I take the trace or the norm down to $\mathbb{Q}$, can I say anything about their congruence properties there? Ideally I'd like a congruence of *something* in the integers.


Thanks!


Edit: Are there any assumptions that you can make that might give congruences mod a prime power? 



Congruences mod primes in Galois extensions
Sure. $a\equiv b\pmod{\mathfrak{P}}$ just means $a-b\in\mathfrak{P}$. Taking norms to any subfield $K$ of $\mathbb{Q}(\zeta\_n)$ (e.g., $\mathbb{Q}$ or $\mathbb{Q}(\zeta\_m)$) gives you $N\_{\mathbb{Q}(\zeta\_n)/K}(a-b)\in N\_{\mathbb{Q(\zeta\_n)}/K}\mathfrak{P}.$ 


For $K=\mathbb{Q}$, the latter norm is just $p^f$ where $f$ is the order of $p\pmod{n}$.


For $K=\mathbb{Q}(\zeta\_m)$, the former norm is $(a-b)^{\phi(n)/\phi(m)}$ and the latter is $\mathfrak{p}^{f'}$, where $f'$ is the easily-calculated relative residue degree. 


This doesn't give you an explicit congruence between $a$ and $b$, but given Gerry's answer, that might have been too much to ask for anyway. On the other hand, if $\phi(n)/\phi(m)$ is small or (as in Alex's answer) if $p$ has few factors in $\mathbb{Q}(\zeta\_m)$, you get something at least slightly non-stupid out.


