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Algebra – Infinite Dihedral Group

I want to prove this without using any of the properties about the field of algebraic numbers (specifically that it is one). Essentially I just want to find a polynomial for which $\cos\frac{2\pi}{n}$ is a root.

I know roots of unity and De Moivre’s theorem is clearly going to be important here but I just can’t see how to actually construct the polynomial from these facts.

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Are you familiar with the Chebyshev polynomials $T_n$ (Wikipedia link)? One property is that

$$T_n(\cos(\theta))=\cos(n\theta)$$

(see this section). Thus

$$T_n(\cos(\tfrac{2\pi}{n}))=\cos(2\pi)=1$$

and therefore an example of a polynomial with $\cos(\frac{2\pi}{n})$ as a root is $T_n-1$.

We know $\cos(2\pi/n)+i\sin(2\pi/n)$ is algebraic ($n$th root of unity). And the complex conjugate of algebraic is algebraic (in fact it’s just another $n$th root of unity), so add it to its conjugate because also the sum of two algebraics is algebraic. Then divide by two.

Let $\theta=\frac{2\pi}{n}$. By De Moivre’s formula we have

$$ (\cos \theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta = 1 $$

Expand the left-hand side using the binomial theorem!

Every other term of the expansion contains a $\pm i$ and an *odd* power of $\sin\theta$; we know these terms cancel out because the imaginary part of the right-hand side is zero.

If we remove all those, what is left is *real* terms with only *even* powers of $\sin \theta$. Substitute $(\sin\theta)^{2n} = (1-\cos^2\theta)^n$. Now you have a polynomial identity with integer coefficients in $\cos \theta$ — in other words, an integer polynomial with $\cos\theta$ as root. Thus $\cos\theta$ is algebraic.

[This is actually one way to derive the Chebyshev polynomials].

Euler’s Formula implies

$$

\left[\cos\left(\frac{2\pi}n\right)+i\sin\left(\frac{2\pi}n\right)\right]^n=1\tag{1}

$$

The Binomial Theorem says

$$

\begin{align}

1

&=\sum_{k=0}^ni^k\binom{n}{k}\cos^{n-k}\!\left(\frac{2\pi}n\right)\sin^k\!\left(\frac{2\pi}n\right)\\

&=\sum_{k=0}^{\lfloor n/2\rfloor}(-1)^k\binom{n}{2k}\cos^{n-2k}\!\left(\frac{2\pi}n\right)\sin^{2k}\!\left(\frac{2\pi}n\right)\\

&+i\sum_{k=0}^{\lfloor(n-1)/2\rfloor}(-1)^k\binom{n}{2k+1}\cos^{n-2k-1}\!\left(\frac{2\pi}n\right)\sin^{2k+1}\!\left(\frac{2\pi}n\right)\tag{2}

\end{align}

$$

Looking at the real part of $(2)$ yields

$$

\begin{align}

1

&=\sum_{k=0}^{\lfloor n/2\rfloor}(-1)^k\binom{n}{2k}\cos^{n-2k}\!\left(\frac{2\pi}n\right)\sin^{2k}\!\left(\frac{2\pi}n\right)\\

&=\sum_{k=0}^{\lfloor n/2\rfloor}(-1)^k\binom{n}{2k}\cos^{n-2k}\!\left(\frac{2\pi}n\right)\left[1-\cos^2\!\left(\frac{2\pi}n\right)\right]^k\\

&=\sum_{m=0}^{\lfloor n/2\rfloor}(-1)^m\cos^{n-2m}\!\left(\frac{2\pi}n\right)\sum_{k=m}^{\lfloor n/2\rfloor}\binom{n}{2k}\binom{k}{m}\tag{3}

\end{align}

$$

Equation $(3)$ gives a polynomial equation whose root is $\cos\!\left(\frac{2\pi}n\right)$. Furthermore, the coefficient of $\cos^n\!\left(\frac{2\pi}n\right)$ is $2^{n-1}$, which implies that $2\cos\!\left(\frac{2\pi}n\right)$ is an algebraic integer. This polynomial also has a root of $1$, so it is reducible.

**Example:** Using $n=7$ in $(3)$, we have that $\cos\!\left(\frac{2\pi}7\right)$ satisfies

$$

64x^7-112x^5+56x^3-7x-1=0\tag{4}

$$

Note that $1$ is also a root of $(4)$.

The fourier transform together with higher frequencies are polynomials in lower frequencies should be enough to write any of them as radicals of polynomials in the other frequencies. And of course that radicals of polynomials are algebraic.

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