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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.

Rhubbarb
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    Roughly speaking, square brackets = "everything you can get by addition, scaling and multiplication", while round brackets = "everything you can get by addition, scaling, multiplication and division". So the latter are being used for fields mostly. There are exceptions, like the notation $A\left(\left(u\right)\right)$ for formal Laurent power series in $u$ over $A$. – darij grinberg Nov 27 '13 at 15:22
  • Indeed, @darijgrinberg, one of my instructors in college made a big deal of the fact that $\mathbb Q(i)=\mathbb Q[i]$. – Lubin Nov 27 '13 at 16:41

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