Let $k=\mathbf Q[i]$ be the field of Gaussian numbers.
I've proved the following easy lemma:


"If $x \in \mathcal{O}\_k$ (the ring of integers of $k$) and $p$ is an odd rational prime dividing the norm $N\_k(x)$, then there is a prime ideal $P$ in $\mathcal{O}\_k$ such that $P$ divides both $p\mathcal{O}\_k$ and $x\mathcal{O}\_k$."


(The proof goes distinguishing if $p \equiv 1 \bmod 4$ or $p \equiv 3 \bmod 4$)


My question is if the same (or a similar) fact is true for other number fields $k$ (?)


Thanks for any idea/reference.



Prime dividing norm of algebraic integer
Yes. It's sufficient to prove $(p,x)$ is not $(1)$, because then we can take any prime factor of $(p,x)$. But if $ap + bx =1 $ for $a,b \in \mathcal O\_K$, then $N(b)N(x) =N(bx) = N(1-ap) \equiv 1$ mod $p$. This is a contradiction as $N(b)$ is an integer if $N(x)$ is divisible by $p$.


