Let $a$ and $b$ be positive integers with GCD $d > 0$. Show that in $\mathbb{Q}[x]$, the ideal generated by $x^a - 1$ and $x^b - 1$ equals the ideal generated by $x^d - 1$.
Here is what I have so far: since $\mathbb{Q}$ is a field, $\mathbb{Q}[x]$ is a Euclidean domain. So $\mathbb{Q}[x]$ is also a PID and $(x^a - 1,x^b - 1) = (\alpha)$ for some $\alpha$. Furthermore we can write $\alpha = p(x)(x^a - 1) + q(x)(x^b - 1)$ for some $p(x), q(x) \in \mathbb{Q}[x]$. Using the norm $N(\alpha) = \text{deg }\alpha$, we have $N(\alpha) = ma + nb$ where $m = \text{deg }p(x)$ and $n = \text{deg }q(x)$. This seems to imply $\text{deg }\alpha = d$, but I am not sure what else to do here.
