Let $\phi$ be a Euclidean function, prove that if $a|b$ and $\phi (a) = \phi (b)$, then $a\sim b$.
So this means that $b=\gamma a$ for some $\gamma$, but beyond this I haven't been able to get everywhere. The quotient-divisor property seems like the thing I need to use but despite playing around with it for a while now I just can't figure out the trick. Essentially I'm trying to show that $\gamma$ is a unit, but so far no luck.
Can anyone help? Thanks.