Suppose that q is a Gaussian or Eisenstein prime and let p be the prime number that lies below q. S is the set of G/E multiples of q. How to prove that S∩Z is the set of integer multiples of p?
I'm unclear of how to use the properties that follow from the fact that p lying below q. Is it q|p that I am supposed to use?
If $p$ is the prime that lies below $q$, then $q\mid p$ or equivalently $p=rq$ for some Gaussian or Eisenstein integer $r$. It follows that all integer multiples of $p$ are in $S$, because $kp=krq\in S$.
Suppose that some $n\in S\cap\Bbb{Z}$ is not an integer multiple of $p$. Then $n=sq\in S$ for some Gaussian or Eisenstein integer $s$, and $p$ and $n$ are coprime so there exist integers $a$ and $b$ such that $ap+bn=1$. It follows that $$1=ap+bn=arq+bsq=(ar+bs)q,$$ which implies that $q$ is a unit. This contradicts the fact that $q$ is a prime. Hence no such integer $n$ exists, and we see that $S\cap\Bbb{Z}=p\Bbb{Z}$.