Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

In lectures we defined the cyclotomic polynomials as $\Phi_n(x):=\prod_{k=1,\gcd(k,n)=1}^n (x-e^{2\pi i k/n})$ and the cyclotomic field $K_n$ as the splitting field of $x^n-1$ over $\mathbb Q$. But we never explicitly talked about the relationship between the two.

Are the following assertions correct?

(1) Let $\xi$ be a primtive $n$-th root of unity, then $K_n=\mathbb Q(\xi)$.

(2) Let $\xi$ be a primtive $n$-th root of unity, then $\Phi_n(x)$ is its minimal polynimial over $\mathbb Q$.

(3) $K_n$ is the splitting field of $\Phi_n(x)$ over $\mathbb Q$.

share|cite|improve this question
Yes, yes and yes. Any more or less decent book in Galois theory deals with this. – DonAntonio Mar 3 '13 at 19:32
up vote 2 down vote accepted

Yes, all statements are correct.

share|cite|improve this answer
Thanks a lot. :) – Thomas Mar 3 '13 at 19:55

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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