Ideals generated by roots of polynomials Let $\alpha$ be a root of $x^3-2x+6$. Let $K=\mathbb{Q}[\alpha]$ and let denote by $\mathscr{O}_K$ the number ring of $K$. Now consider the ideal generated by $(4,\alpha^2,2\alpha,\alpha -3)$ in $\mathscr{O}_K$. I want to prove that this ideal is actually the entire ring $\mathscr{O}_K$. Do i need to know who $\alpha$ is to do this? Is there any theorem which could help?

Hint/steps for one route that caught my eye. There may be a shorter one out there. Call that ideal $I$. Observe that $\alpha \in {\cal O}_K$.


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*Show that $\alpha+1\in I$.

*Show that $\alpha^2-1\in I$.

*Show that $1\in I$.


Remark: this is my second suggestion - the first suggestion had 5 steps.
