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What's the difference between $\mathbb{Q}[\sqrt{-d}]$ and $\mathbb{Q}(\sqrt{-d})$?
Let $$\mathbb Q[i]=\{a+ib|a, b\in \mathbb Q\}$$ Any nonzero element $a+ib\in\mathbb{Q}[i]$ has an inverse element because $$\frac{1}{a+ib}=\frac{a-ib}{a^2+b^2}\in\mathbb{Q}.$$
Is it true that $\mathbb{Q}(i)=\mathbb{Q}[i]$?
Thanks!