Definition of $\mathbb{Z}[\omega]$ where $\omega$ is a primitive root of unity What does $\mathbb{Z}[\omega]$ usually mean when $\omega$ is a primitive root of unity?
 A: $\mathbb{Z}[\omega]$ 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.
A: When $\omega$ is any algebraic element of $\mathbb{C}$ satisfying a monic polynomial with integral coefficients, $\mathbb{Z}[\omega]$ denotes the ring $\{a_0+a_1\omega+\ldots+a_{n-1}\omega^{n-1}: a_i\in \mathbb{Z}, n=\text{deg }\omega\}$, where the degree of $\omega$ is the degree of the minimal polynomial of $\omega$ over $\mathbb{Q}$. When $\omega$ is a primitive $n$-th root of unity, this ring happens to be the ring of integers of the cyclotomic field $\mathbb{Q}(\mu_n)$, but for general algebraic $\omega$, $\mathbb{Z}[\omega]$ is a proper finite index subring of the ring of integers of $\mathbb{Q}[\omega]$.
