What's the difference between $\mathbb{Q}[\sqrt{-d}]$ and $\mathbb{Q}(\sqrt{-d})$? Sorry to ask this, I know it's not really a maths question but a definition question, but Googling didn't help. When asked to show that elements in each are irreducible, is it the same?

In general, if you have a field extension $L/K$, and $\alpha\in L$, we define $K[\alpha]$ to be the subring of $L$ generated by $K$ and $\alpha$, whereas we define $K(\alpha)$ to be the subfield of $L$ generated by $K$ and $\alpha$. Now, it sometimes happens that $K[\alpha]$ is already a field, in which case we have $K[\alpha]=K(\alpha)$. This is what happens with $\mathbb{Q}[\sqrt{-d}]$ and $\mathbb{Q}(\sqrt{-d})$.
