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?

  • 1
    $\begingroup$ Do you know the technique of localization? $\endgroup$
    – Lubin
    Jun 20, 2015 at 2:13
  • $\begingroup$ @Lubin No, I do not. $\endgroup$
    – MathManiac
    Jun 20, 2015 at 4:57

1 Answer 1


This question was previously asked and answered on Math Overflow:

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

Alternatively, a proof can be found here.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.