TITLE: Definition of $\mathbb{Z}[\omega]$ where $\omega$ is a primitive root of unity
QUESTION [5 upvotes]: What does $\mathbb{Z}[\omega]$ usually mean when $\omega$ is a primitive root of unity?

REPLY [6 votes]: $\mathbb{Z}[\omega]$ is the ring generated by $\mathbb{Z}$ and $\omega$ (inside, say $\mathbb{C}$). You can think of this as a ring whose elements are polynomials in $\omega$ with coefficients in $\mathbb{Z}$. Since, $\omega^n=1$ for the appropriate $n$, you need to only consider polynomials of degree less than $n$. Then the usual addition and multiplication of polynomials give you the ring structure.