Minimal subring of complex numbers

Question

Let $\alpha$ be a root of $X^3+X^2-2X+8$.

My question is if $\mathbb Z\left[\alpha,\frac{\alpha+\alpha^2}{2}\right]=\{a+b\alpha +c\frac{\alpha+\alpha^2}{2}:a,b,c\in\mathbb Z\}$?

Thank you all.

Answer 1

Yes. If $\beta = \frac{\alpha + \alpha^2}{2}$, the key fact that you need to verify is that $\alpha^2$, $\alpha \beta$, and $\beta^2$ can all be written in the form $a+b\alpha + c\beta$.

Answer 2

For sure $$\mathbb{Z}[a] \cong \mathbb{Z}[x] / \langle x^3 + x^2 - 2x + 8\rangle \cong_{\text{as } \mathbb{Z} \text{ module}} \langle 1, \alpha, \alpha^2 \rangle$$

If my calculations are correct $(\frac{\alpha + \alpha^2}{2})^2 = \frac{\alpha + \alpha^2}{2} - 2\alpha - 2$. Obviously $\langle 1, \alpha, \alpha^2 \rangle \subseteq \langle 1, \alpha, \frac{\alpha + \alpha^2}{2} \rangle$, so to see if $\mathbb{Z}[\alpha, \frac{\alpha + \alpha^2}{2}] \subseteq \langle 1, \alpha, \frac{\alpha + \alpha^2}{2} \rangle$ you need only to check that $\frac{\alpha + \alpha^2}{2}, \alpha\frac{\alpha + \alpha^2}{2}$ and $\alpha^2\frac{\alpha + \alpha^2}{2}$ are in $\langle 1, \alpha, \frac{\alpha + \alpha^2}{2} \rangle$.