Let $ f(t)=t^{4}-2 t^{3}+2 t^{2}-2 t+1 $, prove that $\mathbb{Q}[t] /(f)$ is a principal ideal ring.
We know that $f(t)$ is the product of two irreducibles:
$$t^{4}-2 t^{3}+2 t^{2}-2 t+1 = (t^2+1)(t^2-2t+1)$$
Hence $\mathbb{Q}[t] /(f)$ is not a field (since $f(t)$ isn't irreducible, it is not maximal in the PIR $\mathbb{Q}[t]$).
How can I conclude that $\mathbb{Q}[t] /(f)$ is not a principal ideal ring?
Hint: is $\mathbb{Q}[t]$ a principal ideal ring? How do ideals of this ring and the ring in question relate?