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

Let $w$ be primitive nth root of unity over $\mathbb{Q}$. Find the minimal polynomial of $w+w^{-1}$ over $\mathbb{Q}$

share|cite|improve this question
Note that $w+w^{-1} = 2 \cos(2\pi/n)$. A minimal polynomial can be recursively found using trigonometric identities: $1=\cos(2\pi) = \cos(n \cdot 2\pi/n) = \dotsc$ – Martin Brandenburg Oct 28 '13 at 14:42

What happens here is similar to what happens with the polynomial $M_w$ of $w$ : there is no general formula, the answers depends on the prime factorization of $n$ (for example, its degree is $\phi(n)$). In general, we only know that $M_w$ divides $X^{n-1}+X^{n-2}+\ldots +1$, and $M_w$ coincides with this polynomial when $n$ is prime.

Similarly, if $n$ is odd say $n=2m+1$, we may define

$$ T_m=\Bigg(\sum_{j=0}^{\lfloor \frac{m}{2} \rfloor} \binom{m-j}{j} (-1)^j x^{m-2j}\Bigg) + \Bigg(\sum_{j=0}^{\lfloor \frac{m-1}{2} \rfloor} \binom{m-1-j}{j} (-1)^j x^{m-1-2j}\Bigg) \tag{1} $$

Then, we have the algebraic identity

$$ T_m\bigg(w+\frac{1}{w}\bigg)=\frac{1}{w^m}\sum_{i=0}^{2m} w^i \tag{2} $$

which shows that $T_m$ annihilates $v=w+\frac{1}{w}$, so the minimal polynomial $M_v$ of $v$ divides $T_m$. When $n$ is an odd prime we have $M_v=T_m$.

share|cite|improve this answer
furthermore, how can i calculate the degree of following extension [$\mathbb{Q}(w+w^{-1}):\mathbb{Q}$] – Le Van Tu Oct 28 '13 at 15:07
@LeVanTu This degree is $\frac{\phi(n)}{2}$ since the degree $[{\mathbb Q}(w):{\mathbb Q}(v)]$ is equal to $2$ ; indeed $w$ satisfies $w^2-vw+1=0$ – Ewan Delanoy Oct 28 '13 at 15:19
thank you.But i wonder how can you determine polynomial $T_m$? – Le Van Tu Oct 28 '13 at 15:34

The minimum polynomial satisfied by $w$ is palindromic. Making the substitution $z=w+w^{-1}$ is a standard method of treating such polynomials, since it divides the degree of the polynomial by half. Look up palindromic polynomials.

share|cite|improve this answer

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.