Suppose we are working in $\Bbb{Q}(\sqrt{-41})$. Given a ideal, for example $(2-\sqrt{-41})$ (we especially work in $\Bbb{Z}[\omega_{-41}]$). We know that this is a Dedekind ring, thus we have unique factorization of ideals. I started to take the norm of this ideal and I get $45$ and this is equal to $5\cdot3\cdot3$. Can I say something like this:
$(2-\sqrt{-41})=(5)(3)(3)=(5,2+\sqrt{-41})(5,2-\sqrt{-41})(3,1+\sqrt{-41})^2(3,1-\sqrt{-41})^2$
The last step is deduced from the fact that the ideals $(3)$ and $(5)$ are not prime. Is this true?! Suppose this is true. Then we can do this also for $(3+\sqrt{-41})$ and we get this result:
$(3+\sqrt{-41})=(5)(5)(2)=(5,2+\sqrt{-41})^2(5,2-\sqrt{-41})^2(2,1+\sqrt{-41})^2$
Here we get the last step from the fact that $-41\equiv3\ \mod\ 4$. Is this all true? But how to do this for $(2-\sqrt{-41},3+\sqrt{-41})$? I hope someone can help me?!