# Finding a principal element in an ideal in a cubic extension

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!

• 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. – Servaes Jan 19 at 0:52
• Could you also give me for $(3, 1 + \sqrt[3]{5})$ (which comes from $(3)$), I have sorted out the others. Thanks! – DesmondMiles Jan 19 at 0:55
• That would be generated by $(2-\sqrt[3]{5})$. – Servaes Jan 19 at 0:56