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I think I'm missing something here but my lecture notes just seem to state that 'clearly' for $x$ and $y$ in the Gaussian integers (elements of $\mathbb{Z}[i]$, a Euclidean domain), if $\nu(x) = \nu(y)$ then clearly $x$ and $y$ are associates. Am I missing something here or is this obvious?

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    So let us wait until the OP dignifies us with an explanation about his/her notation: what is $\,\nu\,$, anyway?2012-05-17

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Either I’m misunderstanding something, or the statement is false.

Let $x=2+i$. The units of $\Bbb Z[i]$ are $1,-1,i$, and $-i$, so the associates of $x$ are $x$, $-2-i$, $-1+2i$, and $1-2i$. Now let $y=1+2i$; $y$ is not an associate of $x$, but $\nu(y)=5=\nu(x)$.

An even better example: $\nu(8+i)=65=\nu(7+4i)$.