So far I am doing the following exercise from the notes "Number rings" written by Peter Stevenhagen (p. 25, #19):
"Let $\tau$ be a zero of an irreducible polynomial $aX^2 + bX + c \in \mathbb{Z}[X]$. Show that $R = \mathbb{Z}[a\tau]$ is an order in the quadratic field $\mathbb{Q}(\sqrt{b^2 − 4ac})$, and that $I = \mathbb{Z}+\mathbb{Z}\tau$ is an invertible $R$-ideal."
There is actually a hint that we can try to compute the ideal $I\sigma(I)$ with $\sigma$ the non-trivial automorphism over $\mathbb{Q}(\sqrt{b^2 − 4ac})$. But since $1$ lies in both $I$ and $\phi(I)$ the product is actually $R$ (which is also the ring of integers of the number field! I've checked this already). This doesn't seem to be right for me. Is my argument correct? Any help is appreciated.