How do I show the ideal $\displaystyle $I = \left( {1 - \sqrt { - 5} } \right)$$ is principal in $\displaystyle ${Z}\left[ {\sqrt { - 5} } \right$$
Thanks!
Right, for example $\displaystyle 3,2+\sqrt{-5},2-\sqrt{-5}$ are irreducible elements in $\displaystyle \mathbb{Z}[\sqrt{-5}]$ and $\displaystyle 9=3\cdot 3=(2+\sqrt{-5})\cdot (2-\sqrt{-5})$ so, $\displaystyle \mathbb{Z}[\sqrt{-5}]$ is not a factorial ring. This example provided by Dedekind has had great importance in the foundation of ideals theory to supply the factorization of elements in not factorial rings.