Show that for $n \geq 2$, the $n^{th}$ cyclotomic polynomial is a reciprocal polynomial, i.e. $\Phi_{n}(x) = x^{\phi(n)}\Phi(n)(x^{-1})$.

Question

Here $\phi(n)$ is the Euler totient function and the degree of $\Phi_{n}(x)$.

What I've done so far: Let $\phi(n) = p$ so the following products each have p components. $$\Phi_{n}(x) = \prod_{k=1, k|n}^n(x - \omega_{k}) = (x - \omega_{1})(x - \omega_{2})...(x - \omega_{p})$$ where $\omega_{k}$ is a primitive $n^{th}$ root of unity.

Then

$$\Phi_{n}(x^{-1}) = \prod_{k=1, k|n}^n(x^{-1} - \omega_{k}) = (x^{-1} - \omega_{1})(x^{-1} - \omega_{2})...(x^{-1} - \omega_{p})$$

and $x^{p} \Phi_{n}(x^{-1}) = x^p(x^{-1} - \omega_{1})(x^{-1} - \omega_{2})...(x^{-1} - \omega_{p})$.

$= x(x^{-1} - \omega_{1})x(x^{-1} - \omega_{2})...x(x^{-1} - \omega_{p})$

(Splitting the $x^p$ amongst the components of the product.)

$= (1 - x\omega_{1})(1 - x\omega_{2})...(1 - x\omega_{p})$.

So now I need to show that

$(x - \omega_{1})(x - \omega_{2})...(x - \omega_{p})= (1 - x\omega_{1})(1 - x\omega_{2})...(1 - x\omega_{p})$.

The LHS has zeros $\omega_{1}, \omega_{1}, ..., \omega_{p}$ and the RHS has zeros $\frac{1}{\omega_{1}}, \frac{1}{\omega_{2}}, ..., \frac{1}{\omega_{p}}$.

If $\omega_{i}$ is an $n^{th}$ root of unity, so is $\omega_{i}^{-1}$, so the zeros of both sides are primitive $n^{th}$ roots of unity. Moreover they have the same number of roots and hence both sides have degree p.

And then... I draw a blank. Suggestions or hints would be much appreciated!

Answer 1

So you know that you are done if you can show $$ (x - \omega_{1})(x - \omega_{2})...(x - \omega_{p})= (1 - x\omega_{1})(1 - x\omega_{2})...(1 - x\omega_{p})$$

which is true if they have the same roots. The LHS has zeros $\omega_i$ and the RHS has zeros $1/\omega_i$ so to finish you have to show that that the second list is a permutation of the first list (in particular, you must show the second list consists of primitive $n$-th roots only, not any $n$-th root as you wrote above). To show this, note that the primitive $n$-th roots of unity are $\exp(2\pi i k/n)$ where $(k,n)=1$ so the inverse is $\exp(- 2\pi i k/n)= \exp(2\pi i (n-k)/n).$ Since $(k,n)=1 \implies (n-k,n)=1$ the inverse is also a primitive $n$-th root of unity.

Answer 2

Note that $$x^n-1=\prod_{d|n}\Phi_d(x),$$ hence by Möbius inversion $$\Phi_n(x)=\prod_{d|n}(x^d-1)^{\mu(d)}. $$ Then the result follows because each $x^d-1$ is reciprocal and products/quotients of reciprocals are reciprocal.

Answer 3

If $n = 2$, then we are talking of the polynomial $x+1$.

If $n > 2$, we have that $1, -1$ are not roots of $\Phi_{n}(x)$, so every root $r$ can be paired with its inverse $r^{-1} \ne r$ to show that the constant coefficient of $\Phi_{n}(x)$, which is the leading coefficient of $x^{\phi(n)}\Phi(n)(x^{-1})$, is $1$.

Also, $x^{\phi(n)}\Phi(n)(x^{-1})$ has the same degree as $\Phi_{n}(x)$, and has the same roots, which are distinct.

It follows that $x^{\phi(n)}\Phi(n)(x^{-1}) = \Phi_{n}(x)$, as the two polynomials are monic, and have the same distinct roots.