<s>[INST] Find all units of $\Bbb Q[x] / (x^2-1)$ Find all units of $\Bbb Q[x] / (x^2-1)$.
I just tried to find them by simply putting $(ax+b)(cx+d)=1$. But it was too complicated to find the undetermined coefficients. I cannot come up with anything related in case the ideal is not irreducible
Could you give me a few hints...?
 [/INST] First, $b+(x^2-1)$ is a unit for all $0\ne b\in\mathbb{Q}$.
Also note that if for $a\ne0$,  $ax+b+(x^2-1)$ is a unit, then so is $x+\frac{b}{a}+(x^2-1)$. So we need to check elements of the form $x+b+(x^2-1)$.
Now:
$$(x+b)(cx+d)=cx^2+dx+bcx+bd=c+bd+(d+bc)x=1$$
i.e. $d=-bc$ and $c+bd=1$ which gives
$$c-b^2c=1\implies c=\frac{1}{1-b^2}\quad\text{and}\quad d=\frac{-b}{1-b^2}$$
Therefore units of $\mathbb{Q}[x]/(x^2-1)$ are of the form $a(x+b)+(x^2-1)$ with $a\ne0$ and $b\ne\pm1$. The inverse is given by
$$(a(x+b)+(x^2-1))^{-1}=\frac{1}{a}\frac{1}{1-b^2}(x-b)+(x^2-1)$$
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