# Ring of integers of a cyclotomic number field

Let $\omega$ be the primitive $n^{th}$ root of unity. Consider the number field $\mathbb Q(\omega)$. How to show that the ring of integers for this field is $\mathbb Z(\omega)$?

Also, find the discriminant of $\mathbb Z(\omega)/\mathbb Z$.

If $n$ is a prime, then finding the discriminant is easy using the concept of norm. But how to do it in a general case?

• Do you know the technique of localization? Jun 20, 2015 at 2:13
• @Lubin No, I do not. Jun 20, 2015 at 4:57

Quick proof of the fact that the ring of integers of $\mathbb{Q}[\zeta_n]$ is $\mathbb{Z}[\zeta_n]$?