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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?

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1 Answer

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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.

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