Find a non-principal ideal in $ \Bbb Z [2i]$.

Question

Find a non-principal ideal in $ \Bbb Z [2i]$.

I think it might be $(1+2i,1-2i)$, but have problems with proving this.

I know that $|1+2i|=|1-2i|=5$.

Moreover, there are only 6 elements with non-bigger norm than 5 (except these two), I mean $1,-1,2,-2,2i,-2i$. None of these has norm equal to 5, so if this ring was principal, then $1+2i = a(1-2i)$, where $|a|=1$. Of course this is not possible, so our ideal is not principal.

Is this correct?

Answer 1

Hint. $\mathbb Z[2i]\simeq\mathbb Z[X]/(X^2+4)$. Now recall the usual example of a non-principal ideal in $\mathbb Z[X]$.

$(2,2i)$ is a non-principal ideal in $\mathbb Z[2i]$.