For example the notation for rings like the Gaussian integers uses brackets: $\mathbb{Z}[i]$.
Yet the notation for field extensions uses parentheses, like in the case of the Gaussian rationals $\mathbb{Q}(i)$.
But why?
In both cases we have $$ \mathbb{Z}[i] = \mathbb{Z}[X]/(X^2 + 1) \\ \mathbb{Q}[i] = \mathbb{Q}[X]/(X^2 + 1) \, .$$
Is it just convention, or is there any deeper meaning that I'm missing?
Typically the notation $R[\alpha]$ means that we are adjoining the element $\alpha$ to some a ring $R$. However, $R(\alpha)$ means that we are adjoining the element $\alpha$ and $\alpha^{-1}$ to the ring $R$. But when we are adjoining roots of polynomials, we get inverses for free, because if we are adjoining the root $r$ of some polynomial $p(z) = a_nz^n +.... + a_0$, then $$ r(a_n r^{n-1} + ... + a_1) = - a_0. $$ Typically, $-a_0$ will be invertible in our ring, and so $r$ is invertible. One specific instance of this is that $\mathbb{Q}(i) = \mathbb{Q}[i]$.
A case where they would not be the same is $\mathbb{Q}[t]$ and $\mathbb{Q}(t)$, where $t$ is a transcendental element over $\mathbb{Q}$. The former would just be isomorphic to the ring of polynomials with $\mathbb{Q}$ coefficients, which is not a field. The latter would be the field of rational functions with $\mathbb{Q}$ coefficients.
While I'm not sure if this reasoning is standard, I was exposed to this idea by an algebra professor of mine. I trust his knowledge of the conventions a bit better than my own.