What do the Squared Brackets stand for in $\mathbb{Z}[\mathrm{i}] $

Question

What do the Squared Brackets stand for in e.g. $\mathbb{Z}[\mathrm{i}] $?

Answer 1

In general, if you have two rings $P\subseteq Q$ and an element $q$ of $Q$, then $P[q]$ denotes the smallest subring of $Q$ which contains every element of $P$ and additionally $q$.

In this particular example, we have $P=\Bbb Z,Q=\Bbb C,q=i$, so $\Bbb Z[i]$ is the smallest ring of complex numbers which contains all the integers and imaginary unit $i$. Because $i^2=-1$, it is easy to see that this subring is precisely the set of numbers of the form $a+bi$ for $a,b\in\Bbb Z$.

Answer 2

In general, the ring $R[x]$ refers to polynomials on $x$ with coefficients in $R$. Applying this to your example, $\Bbb Z[i]$ consists of elements of the form $\sum_{j = 0}^n a_j i^j$, where the $a_j$ are integers. Note, however, that since $i^2 = -1$, such an element can always be rewritten in the form $a+bi$.

Answer 3

This refers to the Gaussian integers, or all of the complex numbers $a+bi$ for $a, b \in \mathbb{Z}$. Basically, it's the ring generated by the $\mathbb{Z}$ with the element $i$ included. I think that is why the notation of $\mathbb{Z}[i]$ is used.