Showing that an ideal in $\mathbb{Z}[\sqrt{-21}]$ is principal

Question

I have $\mathfrak{a}=(5,\sqrt{-21}-2).$ Can anyone tell me why $\mathfrak{a}^2$ is principal? I have multiplied the ideals out to obtain $$(5, 5\sqrt{-21}-10,-17-4\sqrt{-21}) $$ How does this reduce to a principal ideal?

Answer 1

The ideal $\mathfrak{a}^2$ is the ideal generated by the products of the generators of $\mathfrak{a}$, hence $$\mathfrak{a}^2=(5^2,5(\sqrt{-21}-2),(\sqrt{-21}-2)^2)=(25,-10+5\sqrt{-21},-17-4\sqrt{-21}).$$ The latter two generators are clearly both multiples of $\sqrt{-21}-2$, and the identity $$-(\sqrt{-21}+2)(\sqrt{-21}-2)=25,$$ shows that the first generator is also a multiple of $\sqrt{-21}-2$.