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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?

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# Congruences mod primes in Galois extensions

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!