Z denotes the set of integers, but what is $Z[i]$

Question

I came across $Z[i]$ in a few questions. I know Z denotes the set of integers but what does $Z[i]$ denote? Is it similar to the array mechanism where i represents the position of an integer in the set Z?

Answer 1

The set $S[x]$ is the polynomials in $x$ with coefficients in $S$. Since $i^2+1=0$, $\mathbb{R}[i]$ is the complex numbers. Similarly, $\mathbb{Z}[i]=\{ a+bi|a,\,b\in \mathbb{Z}\}$.