Is there a reason for the difference in the use of parentheses versus brackets as used in algebraic extensions. For example, when the field rational numbers ${\mathbb{Q}}$ extended with $i = \sqrt{-1}$ is denoted ${\mathbb{Q}}(i)$, whereas the ring of integers ${\mathbb{Z}}$ extended with $i$ is denoted ${\mathbb{Z}}[i]$?
In particular, does this difference in notation in itself carry any meaning beyond the applications to rings versus fields, for example? Could, for example, '${\mathbb{Q}}[i]$' and '${\mathbb{Z}}(i)$' (or similar) have useful, but different, meanings?
Brackets are also used in the context of rings of polynomials (such as ${\mathbb{Z}}[X]$, ${\mathbb{Q}}[X]$ and ${\mathbb{R}}[X]$, etc.). Is this related?
Thanks.