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I am struggling with how to find a principal element for each of $(2, 1 + \sqrt[3]{5})$ and $(2, \sqrt[3]{25} - \sqrt[3]{5} + 1)$ in $\mathbb{Z}[\sqrt[3]{5}]$. Working with norms here seems ugly. The main aim is to prove it is a PID via factoring $(2)$, $(3)$, $(5)$, $(7)$ and the Minkowski bound.

Any help appreciated!

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    $\begingroup$ Well one way would be to use sage to find $(\sqrt[3]{25}+2\sqrt[3]{5}+3)$ for the first ideal and $(\sqrt[3]{25}-\sqrt[3]{5}+1)$ for the second. $\endgroup$ – Servaes Jan 19 at 0:52
  • $\begingroup$ Could you also give me for $(3, 1 + \sqrt[3]{5})$ (which comes from $(3)$), I have sorted out the others. Thanks! $\endgroup$ – DesmondMiles Jan 19 at 0:55
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    $\begingroup$ That would be generated by $(2-\sqrt[3]{5})$. $\endgroup$ – Servaes Jan 19 at 0:56

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