Let $K$ be a number field. Let $\frak p$ be a prime ideal in $\mathcal O_K$. Let $u\in \mathcal O_K$ and $m\in \mathbb N$. I've been told that $|u|_{\frak p} = |m|_{\frak p} = 1$ where $|\cdot|_{\frak p}$ denotes the $\frak p$-adic norm. Why should this be true?
Given the definition of $|\cdot|_{\frak p}$, this is equivalent to saying that when we factor $u\mathcal{O_K}$ and $m\mathcal{O_K}$ into prime ideals, $\frak p$ has multiplicity 0, but I still don't see why this should be true.
Without additional conditions on (say) $u$ and $m$, this is certainly false. For example, if $\mathfrak{p}$ is prime in $\mathcal{O}_K$, let $u$ be any nonzero element of $\mathfrak{p}$. Then $(u)\subseteq \mathfrak{p}$, so that $|u|_{\mathfrak{p}}>0$. For $m$, if $\mathfrak{p}$ lies over the rational prime $p$, let $m=p$. Then $|m|_{\mathfrak{p}}>0$.