Still going over Alaca & Williams (I might die before I fully understand that book).
In $\mathbb{Z}[\sqrt{-21}]$, the factorization of $\langle 2 \rangle$ is $\langle 2, 1 + \sqrt{-21} \rangle^2$. I can easily see that this ideal contains both the number $1 + \sqrt{-21}$ and the number $1 - \sqrt{-21}$.
In $\mathbb{Z}[\sqrt{23}]$, the factorization of $\langle 2 \rangle$ is $\langle 2, 1 + \sqrt{23} \rangle^2$. But $\mathbb{Z}[\sqrt{23}]$ is a principal ideal domain, which means that $\langle 2, 1 + \sqrt{23} \rangle$ is generated by a single number in this domain. Since $(5 - \sqrt{23})(5 + \sqrt{23}) = 2$, I think that number is $5 + \sqrt{23}$.
But the problem that I'm having is that I don't see how $5 - \sqrt{23}$ is contained in $\langle 5 + \sqrt{23} \rangle$. I tried $$\frac{5 + \sqrt{23}}{5 - \sqrt{23}}$$ on my scientific calculator, but I don't recognize what this number (approximately $0.020842$) could be in this ring.