An example showing $\mathbb{Z}[\sqrt[3]{7}]$ is not a UFD [closed]
Solution 1:
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Take $\mathfrak{p}$ a non-principal inversible prime ideal.
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Take $a\in \mathfrak{p},\not \in \mathfrak{p}^2$ (this can be done in the finite ring $O_K/(p^2)$ where $p=char(O_K/\mathfrak{p})$)
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Let $m$ be the order of $\mathfrak{p}$ in the class group, so $\mathfrak{p}^m=(b)$.
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$b$ is irreducible but $(b)$ is not a prime ideal.
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$a^m \in (b)$
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Neither $a$ nor $a^{m-1}$ is in $(b)$ so $a^m$ has more than one factorization.