For an integer ${\displaystyle n\geq 1}$, the minimal polynomial ${\displaystyle \Psi _{n}(x)}$ of ${\displaystyle 2\cos \left({\frac {2\pi }{n}}\right)}$ is the non-zero monic polynomial of degree ${\displaystyle 1}$ for ${\displaystyle n=1,2}$ and degree ${\displaystyle {\frac {1}{2}}\varphi (n)}$ for ${\displaystyle n\geq 3}$ with integer coefficients, such that ${\displaystyle \Psi _{n}\!\left(2\cos \left({\frac {2\pi }{n}}\right)\right)=0}$. Here ${\displaystyle \varphi (n)}$ denotes the Euler's totient function. In particular, for ${\displaystyle n\leq 2,}$ one has ${\displaystyle \Psi _{1}(x)=x-2}$ and ${\displaystyle \Psi _{2}(x)=x+2.}$

For every n, the polynomial ${\displaystyle \Psi _{n}(x)}$ is monic, has integer coefficients, and is irreducible over the integers and the rational numbers. All its roots are real; they are the real numbers ${\displaystyle 2\cos \left({\frac {2k\pi }{n}}\right)}$ with k coprime with n and 1 ≤ k ≤ n (coprimality implies that k = n can occur only for n = 1). These roots are twice the real parts of the primitive nth roots of unity.

The polynomials ${\displaystyle \Psi _{n}(x)}$ are typical examples of irreducible polynomials whose roots are all real and which have a cyclic Galois group.

Examples

The first few polynomials ${\displaystyle \Psi _{n}(x)}$ are

${\displaystyle {\begin{aligned}\Psi _{1}(x)&=x-2\\\Psi _{2}(x)&=x+2\\\Psi _{3}(x)&=x+1\\\Psi _{4}(x)&=x\\\Psi _{5}(x)&=x^{2}+x-1\\\Psi _{6}(x)&=x-1\\\Psi _{7}(x)&=x^{3}+x^{2}-2x-1\\\Psi _{8}(x)&=x^{2}-2\\\Psi _{9}(x)&=x^{3}-3x+1\\\Psi _{10}(x)&=x^{2}-x-1\\\Psi _{11}(x)&=x^{5}+x^{4}-4x^{3}-3x^{2}+3x+1\\\Psi _{12}(x)&=x^{2}-3\\\Psi _{13}(x)&=x^{6}+x^{5}-5x^{4}-4x^{3}+6x^{2}+3x-1\\\Psi _{14}(x)&=x^{3}-x^{2}-2x+1\\\Psi _{15}(x)&=x^{4}-x^{3}-4x^{2}+4x+1\\\Psi _{16}(x)&=x^{4}-4x^{2}+2\\\Psi _{17}(x)&=x^{8}+x^{7}-7x^{6}-6x^{5}+15x^{4}+10x^{3}-10x^{2}-4x+1\\\Psi _{18}(x)&=x^{3}-3x-1\\\Psi _{19}(x)&=x^{9}+x^{8}-8x^{7}-7x^{6}+21x^{5}+15x^{4}-20x^{3}-10x^{2}+5x+1\\\Psi _{20}(x)&=x^{4}-5x^{2}+5\end{aligned}}}$

Explicit form if n is odd

If ${\displaystyle n}$ is an odd prime, the polynomial ${\displaystyle \Psi _{n}(x)}$ can be written in terms of binomial coefficients following a "zigzag path" through Pascal's triangle:

Putting ${\displaystyle n=2m+1}$ and

${\displaystyle {\begin{aligned}\chi _{n}(x):&={\binom {m}{0}}x^{m}+{\binom {m-1}{0}}x^{m-1}-{\binom {m-1}{1}}x^{m-2}-{\binom {m-2}{1}}x^{m-3}+{\binom {m-2}{2}}x^{m-4}+{\binom {m-3}{2}}x^{m-5}--++\cdots \\&=\sum _{k=0}^{m}(-1)^{\lfloor k/2\rfloor }{\binom {m-\lfloor (k+1)/2\rfloor }{\lfloor k/2\rfloor }}x^{m-k}\\&={\binom {m}{m}}x^{m}+{\binom {m-1}{m-1}}x^{m-1}-{\binom {m-1}{m-2}}x^{m-2}-{\binom {m-2}{m-3}}x^{m-3}+{\binom {m-2}{m-4}}x^{m-4}+{\binom {m-3}{m-5}}x^{m-5}--++\cdots \\&=\sum _{k=0}^{m}(-1)^{\lfloor (m-k)/2\rfloor }{\binom {\lfloor (m+k)/2\rfloor }{k}}x^{k},\end{aligned}}}$

then we have ${\displaystyle \Psi _{p}(x)=\chi _{p}(x)}$ for primes ${\displaystyle p}$.

If ${\displaystyle n}$ is odd but not a prime, the same polynomial ${\displaystyle \chi _{n}(x)}$, as can be expected, is reducible and, corresponding to the structure of the cyclotomic polynomials ${\displaystyle \Phi _{d}(x)}$ reflected by the formula ${\displaystyle \prod _{d\mid n}\Phi _{d}(x)=x^{n}-1}$, turns out to be just the product of all ${\displaystyle \Psi _{d}(x)}$ for the divisors ${\displaystyle d>1}$ of ${\displaystyle n}$, including ${\displaystyle n}$ itself:

${\displaystyle \prod _{d\mid n \atop d>1}\Psi _{d}(x)=\chi _{n}(x).}$

This means that the ${\displaystyle \Psi _{d}(x)}$ are exactly the irreducible factors of ${\displaystyle \chi _{n}(x)}$, which allows to easily obtain ${\displaystyle \Psi _{d}(x)}$ for any odd ${\displaystyle d}$, knowing its degree ${\displaystyle {\frac {1}{2}}\varphi (d)}$. For example,

${\displaystyle {\begin{aligned}\chi _{15}(x)&=x^{7}+x^{6}-6x^{5}-5x^{4}+10x^{3}+6x^{2}-4x-1\\&=(x+1)(x^{2}+x-1)(x^{4}-x^{3}-4x^{2}+4x+1)\\&=\Psi _{3}(x)\cdot \Psi _{5}(x)\cdot \Psi _{15}(x).\end{aligned}}}$

Explicit form if n is even

From the below formula in terms of Chebyshev polynomials and the product formula for odd ${\displaystyle n}$ above, we can derive for even ${\displaystyle n}$

${\displaystyle \prod _{d\mid n \atop d>1}\Psi _{d}(x)={\Big (}\chi _{n+1}(x)+\chi _{n-1}(x){\Big )}.}$

Independently of this, if ${\displaystyle n=2^{k}}$ is an even prime power, we have for ${\displaystyle k\geq 2}$ the recursion (see OEIS: A158982)

${\displaystyle \Psi _{2^{k+1}}(x)=(\Psi _{2^{k}}(x))^{2}-2}$,

starting with ${\displaystyle \Psi _{4}(x)=x}$.

Roots

The roots of ${\displaystyle \Psi _{n}(x)}$ are given by ${\displaystyle 2\cos \left({\frac {2\pi k}{n}}\right)}$,^[1] where ${\displaystyle 1\leq k<{\frac {n}{2}}}$ and ${\displaystyle \gcd(k,n)=1}$. Since ${\displaystyle \Psi _{n}(x)}$ is monic, we have

${\displaystyle \Psi _{n}(x)=\displaystyle \prod _{\begin{array}{c}1\leq k<{\frac {n}{2}}\\\gcd(k,n)=1\end{array}}\left(x-2\cos \left({\frac {2\pi k}{n}}\right)\right).}$

Combining this result with the fact that the function ${\displaystyle \cos(x)}$ is even, we find that ${\displaystyle 2\cos \left({\frac {2\pi k}{n}}\right)}$ is an algebraic integer for any positive integer ${\displaystyle n}$ and any integer ${\displaystyle k}$.

Relation to the cyclotomic polynomials

For a positive integer ${\displaystyle n}$, let ${\displaystyle \zeta _{n}=\exp \left({\frac {2\pi i}{n}}\right)=\cos \left({\frac {2\pi }{n}}\right)+\sin \left({\frac {2\pi }{n}}\right)i}$, a primitive ${\displaystyle n}$-th root of unity. Then the minimal polynomial of ${\displaystyle \zeta _{n}}$ is given by the ${\displaystyle n}$-th cyclotomic polynomial ${\displaystyle \Phi _{n}(x)}$. Since ${\displaystyle \zeta _{n}^{-1}=\cos \left({\frac {2\pi }{n}}\right)-\sin \left({\frac {2\pi }{n}}\right)i}$, the relation between ${\displaystyle 2\cos \left({\frac {2\pi }{n}}\right)}$ and ${\displaystyle \zeta _{n}}$ is given by ${\displaystyle 2\cos \left({\frac {2\pi }{n}}\right)=\zeta _{n}+\zeta _{n}^{-1}}$. This relation can be exhibited in the following identity proved by Lehmer, which holds for any non-zero complex number ${\displaystyle z}$:^[2]

${\displaystyle \Psi _{n}\left(z+z^{-1}\right)=z^{-{\frac {\varphi (n)}{2}}}\Phi _{n}(z)}$

Relation to Chebyshev polynomials

In 1993, Watkins and Zeitlin established the following relation between ${\displaystyle \Psi _{n}(x)}$ and Chebyshev polynomials of the first kind.^[1]

If ${\displaystyle n=2s+1}$ is odd, then

${\displaystyle \prod _{d\mid n}\Psi _{d}(2x)=2(T_{s+1}(x)-T_{s}(x)),}$

and if ${\displaystyle n=2s}$ is even, then

${\displaystyle \prod _{d\mid n}\Psi _{d}(2x)=2(T_{s+1}(x)-T_{s-1}(x)).}$

If ${\displaystyle n}$ is a power of ${\displaystyle 2}$, we have moreover directly^[3]

${\displaystyle \Psi _{2^{k+1}}(2x)=2T_{2^{k-1}}(x).}$

Absolute value of the constant coefficient

The absolute value of the constant coefficient of ${\displaystyle \Psi _{n}(x)}$ can be determined as follows:^[4]

${\displaystyle |\Psi _{n}(0)|={\begin{cases}0&{\text{if}}\ n=4,\\2&{\text{if}}\ n=2^{k},k\geq 0,k\neq 2,\\p&{\text{if}}\ n=4p^{k},k\geq 1,p>2\ {\text{prime,}}\\1&{\text{otherwise.}}\end{cases}}}$

Generated algebraic number field

The algebraic number field ${\displaystyle K_{n}=\mathbb {Q} \left(\zeta _{n}+\zeta _{n}^{-1}\right)}$ is the maximal real subfield of a cyclotomic field ${\displaystyle \mathbb {Q} (\zeta _{n})}$. If ${\displaystyle {\mathcal {O}}_{K_{n}}}$ denotes the ring of integers of ${\displaystyle K_{n}}$, then ${\displaystyle {\mathcal {O}}_{K_{n}}=\mathbb {Z} \left[\zeta _{n}+\zeta _{n}^{-1}\right]}$. In other words, the set ${\displaystyle \left\{1,\zeta _{n}+\zeta _{n}^{-1},\ldots ,\left(\zeta _{n}+\zeta _{n}^{-1}\right)^{{\frac {\varphi (n)}{2}}-1}\right\}}$ is an integral basis of ${\displaystyle {\mathcal {O}}_{K_{n}}}$. In view of this, the discriminant of the algebraic number field ${\displaystyle K_{n}}$ is equal to the discriminant of the polynomial ${\displaystyle \Psi _{n}(x)}$, that is^[5]

${\displaystyle D_{K_{n}}={\begin{cases}2^{(m-1)2^{m-2}-1}&{\text{if}}\ n=2^{m},m>2,\\p^{(mp^{m}-(m+1)p^{m-1}-1)/2}&{\text{if}}\ n=p^{m}\ {\text{or}}\ 2p^{m},p>2\ {\text{prime}},\\\left(\prod _{i=1}^{\omega (n)}p_{i}^{e_{i}-1/(p_{i}-1)}\right)^{\frac {\varphi (n)}{2}}&{\text{if}}\ \omega (n)>1,k\neq 2p^{m}.\end{cases}}}$

References