Express elements of $\mathbb{Q}(\alpha) $ as polynomials of degree at most 2 with coefficients in $\mathbb{Q}$

Question

Express $\alpha^4 + 2\alpha^5$, $\frac{\alpha}{\alpha - 1} \in \mathbb{Q}(\alpha) $ as polynomials of degree at most 2 with coefficients in $\mathbb{Q}$

where $\alpha = \sqrt[3]{7}$

For the first one I have $14\alpha^2 + 7\alpha$ but I'm not really sure what I'm doing .. any help is appreciated

Answer 1

For the second one, you can multiply the numerator and denominator by $\alpha^2+\alpha+1$ $$\therefore \frac{\alpha(\alpha^2+\alpha+1)}{\alpha^3-1}$$ $\because \alpha^3=7$, we get $$\frac{\alpha^2+\alpha+7}{6}$$

Is this what you wanted?