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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.

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    Well in general $\phi (nm)\geq \phi (n)$, for any integral domain on which a euclidean function can be defined.2012-12-18

1 Answers 1

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Write $a=bq+r$

Then

$a|a, a|b \Rightarrow a|r \,.$

From here, you should get that $r =0$, otherwise you have $a|r$ and $\phi(r)< \phi(b)=\phi(a)$.