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This is an assumption made in Marcus, Number Fields.

$\textbf Z[\omega]$ is UFD for $\omega$ $p-$th root of unity with $3 < p < 23$.

For $p = 2, 3$, one can use Neukrich geometric proof on the distance to the nearest lattice point. It seems that this proof fails for $p=5$ as there are lattice points which is not generated by shifting the fundamental lattice $0,\omega,i\omega,(1+i)\omega$.

Q: Was there a unified approach to deal with this problem for $3 < p < 23$?

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    $\begingroup$ The most straightforward way is to compute Minkowski's bound and factor all primes below it into prime ideals. If all of them are principal, the number ring is a UFD (note that this still forces you to do separate calculations for each $p$). If you keep reading Marcus' book, you will learn more about this approach. $\endgroup$ – Wojowu Oct 8 '17 at 21:01
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    $\begingroup$ This is a very minor detail which I could be wrong about: shouldn't that say "$p$-th root" instead of $p-$th root"? $\endgroup$ – Robert Soupe Oct 10 '17 at 3:33

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