Here is a question in my homework.
Let $\alpha$ be a root of $X^3+X^2-2X+1\in\mathbb{Q}[X]$. Express $(1-\alpha^2)^{-1}$ as a $\mathbb{Q}$-linear combination of $1$, $\alpha$ and $\alpha^2$. Justify the assertion that the cubic is irreducible over $\mathbb{Q}$, using Gauss' Lemma.
This is the first question of my homework so I kind of expect a fast solution. But I couldn't do it the first part.
For the last part, I showed that this is irreducible over $\mathbb{Z}$ because it is cubic and has no solution in $\mathbb{Z}$ and applied Gauss' lemma. If someone kindly shows me how to do the first part, can you please also say why this relates to the second part?