Originally Posted by

**Carl** Hi there

I have the ring $\displaystyle \mathbb{Z}[\sqrt{-5}]$ with field of fractions $\displaystyle K$ and am to decide whether any of the ideals $\displaystyle (2), (1+\sqrt{-5}), (2, 1+\sqrt{-5})$ are invertible or not.

Well, if the ideal $\displaystyle I$ is invertible, we have that $\displaystyle II^*=R$ where $\displaystyle I^*=\{q\in K | qI \subset R\}$

so $\displaystyle q$ is on the form $\displaystyle \frac{a+b\sqrt{-5}}{c+d\sqrt{-5}}$, correct?

Then I would say that the ideal $\displaystyle (2, 1+\sqrt{-5})$ is invertible, because whatever the denominator is in the above fraction, it disappaers when multiplying with $\displaystyle (2, 1+\sqrt{-5})$.

Did I get that right?