View source for Minimal polynomial of 2cos(2pi/n)

{{Short description|Equation for the real part of a root of unity}}
In [[number theory]], the real parts of the [[primitive root of unity|roots of unity]] are related to one-another by means of the [[Minimal polynomial (field theory)|minimal polynomial]] of <math>2\cos(2\pi/n).</math> The roots of the minimal polynomial are twice the real part of the roots of unity, where the real part of a root of unity is just <math>\cos\left(2k\pi/n\right)</math> with <math>k</math> [[coprime]] with <math>n.</math>

==Formal definition==
For an [[integer]] <math>n \geq 1</math>, the [[Minimal polynomial (field theory)|minimal polynomial]] <math>\Psi_n(x)</math> of <math>2\cos(2\pi/n)</math> is the non-zero [[monic polynomial]] of smallest degree for which <math>\Psi_n\!\left(2\cos(2\pi/n)\right) = 0</math>.

For every {{mvar|n}}, the polynomial <math>\Psi_n(x)</math> is monic, has integer coefficients, and is [[irreducible polynomial|irreducible]] over the integers and the [[rational number]]s. All its [[zero of a function|roots]] are [[real number|real]]; they are the real numbers <math>2\cos\left(2k\pi/n\right)</math> with <math>k</math> [[coprime]] with <math>n</math> and either <math>1 \le k < n</math> or <math>k=n=1.</math> These roots are twice the [[real part]]s of the [[primitive root of unity|primitive {{mvar|n}}th roots of unity]]. The number of integers <math>k</math> relatively prime to <math>n</math> is given by [[Euler's totient function]] <math>\varphi(n);</math> it follows that the [[degree of a polynomial|degree]] of <math>\Psi_n(x)</math> is <math>1</math> for <math>n=1,2</math> and <math>\varphi(n)/2</math> for <math>n\geq 3.</math>

The first two polynomials are <math>\Psi_1(x) = x - 2</math> and <math>\Psi_2(x) = x + 2.</math>

The polynomials <math>\Psi_n(x)</math> are typical examples of irreducible polynomials whose roots are all real and which have a [[cyclic group|cyclic]] [[Galois group]].

== Examples ==

The first few polynomials <math>\Psi_n(x)</math> are

: <math> \begin{align}
\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 - 5 x^4 - 4 x^3 + 6 x^2 + 3 x - 1\\
\Psi_{14}(x) &= x^3 - x^2 - 2 x + 1 \\
\Psi_{15}(x) &= x^4 - x^3 - 4 x^2 + 4 x + 1 \\
\Psi_{16}(x) &= x^4 - 4 x^2 + 2 \\
\Psi_{17}(x) &= x^8 + x^7 - 7 x^6 - 6 x^5 + 15 x^4 + 10 x^3 - 10 x^2 - 4 x + 1 \\
\Psi_{18}(x) &= x^3 - 3 x - 1 \\
\Psi_{19}(x) &= x^9 + x^8 - 8 x^7 - 7 x^6 + 21 x^5 + 15 x^4 - 20 x^3 - 10 x^2 + 5 x + 1 \\
\Psi_{20}(x) &= x^4 - 5 x^2 + 5
\end{align}</math>

== Explicit form if ''n'' is odd ==
If <math>n</math> is an odd prime, the polynomial <math>\Psi_{n}(x)</math> can be written in terms of binomial coefficients following a "zigzag path" through [[Pascal's triangle]]:  

Putting <math>n=2m+1</math> and
:<math>\begin{align}
\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{align}</math>
then we have <math>\Psi_{p}(x)=\chi_{p}(x)</math> for primes <math>p</math>.  

If <math>n</math> is odd but not a prime, the same polynomial <math>\chi_{n}(x)</math>, as can be expected, is reducible and, corresponding to the structure of the [[cyclotomic polynomials]] <math>\Phi_{d}(x)</math> reflected by the formula <math>\prod_{d\mid n}\Phi_d(x) = x^n - 1</math>, turns out to be just the product of all <math>\Psi_{d}(x)</math> for the divisors <math>d>1</math> of <math>n</math>, including  <math>n</math> itself: 

:<math>\prod _{d\mid n\atop d>1}\Psi_{d}( x)=\chi_{n}( x).</math>
This means that the <math>\Psi_{d}(x)</math> are exactly the irreducible factors of <math>\chi_{n}(x)</math>, which allows to easily obtain <math>\Psi_{d}(x)</math> for any odd <math>d</math>, knowing its degree <math>\frac{1}{2}\varphi(d)</math>. For example, 
:<math>\begin{align}
\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{align}</math>

== Explicit form if ''n'' is even ==
From the below [[Minimal_polynomial_of_2cos(2pi/n)#Relation_to_Chebyshev_polynomials|formula in terms of Chebyshev polynomials]] and the product formula for odd <math>n</math> above, we can derive for even <math>n</math>

:<math>\prod _{d\mid n\atop d>1}\Psi_{d}( x)= \Big(\chi_{n+1}(x)+\chi_{n-1}(x)\Big).</math>

Independently of this, if <math>n=2^k</math> is an even prime power, we have for <math>k\ge 2</math> the recursion (see {{OEIS2C|id=A158982}})
:<math>\Psi_{2^{k+1}}( x)=(\Psi_{2^k}( x))^2-2</math>,
starting with <math>\Psi_{4}( x)=x</math>.

== Roots ==
The roots of <math>\Psi_n(x)</math> are given by <math>2\cos\left(\frac{2\pi k}{n}\right)</math>,<ref name="watkins-zetilin">{{cite journal|author=W. Watkins and J. Zeitlin|year=1993|title=The minimal polynomial of <math>\cos(2\pi/n)</math>|journal=[[The American Mathematical Monthly]]|volume=100|issue=5|pages=471–474|doi=10.2307/2324301|jstor=2324301|doi-access=free}}</ref> where <math>1 \leq k < \frac{n}{2}</math> and <math>\gcd(k, n) = 1</math>. Since <math>\Psi_n(x)</math> is monic, we have

: <math>\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).</math>

Combining this result with the fact that the [[function (mathematics)|function]] <math>\cos(x)</math> is [[even function|even]], we find that <math>2\cos\left(\frac{2\pi k}{n}\right)</math> is an [[algebraic integer]] for any positive integer <math>n</math> and any integer <math>k</math>.

== Relation to the cyclotomic polynomials ==
For a positive integer <math>n</math>, let <math>\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</math>, a primitive <math>n</math>-th root of unity. Then the minimal polynomial of <math>\zeta_n</math> is given by the <math>n</math>-th [[cyclotomic polynomial]] <math>\Phi_n(x)</math>. Since <math>\zeta_n^{-1} = \cos\left(\frac{2\pi}{n}\right) - \sin\left(\frac{2\pi}{n}\right)i</math>, the relation between <math>2\cos\left(\frac{2\pi}{n}\right)</math> and <math>\zeta_n</math> is given by <math>2\cos\left(\frac{2\pi}{n}\right) = \zeta_n + \zeta_n^{-1}</math>. This relation can be exhibited in the following identity proved by Lehmer, which holds for any non-zero [[complex number]] <math>z</math>:<ref>{{cite journal|author=D. H. Lehmer|year=1933|title=A note on trigonometric algebraic numbers|journal=[[The American Mathematical Monthly]]|volume=40|issue=3|pages=165–166|doi=10.2307/2301023|jstor=2301023}}</ref>

: <math>\Psi_n\left(z + z^{-1}\right) = z^{-\frac{\varphi(n)}{2}}\Phi_n(z)</math>

== Relation to Chebyshev polynomials ==

In 1993, Watkins and Zeitlin established the following relation between <math>\Psi_n(x)</math> and [[Chebyshev polynomials]] of the first kind.<ref name="watkins-zetilin" />

If <math>n = 2s + 1</math> is [[parity (mathematics)|odd]], then{{check source|date=June 2021}}

: <math>\prod_{d \mid n}\Psi_d(2x) = 2(T_{s + 1}(x) - T_s(x)),</math>

and if <math>n = 2s</math> is [[parity (mathematics)|even]], then

: <math>\prod_{d \mid n}\Psi_d(2x) = 2(T_{s + 1}(x) - T_{s - 1}(x)).</math>

If <math>n </math> is a power of <math>2</math>, we have moreover directly<ref>{{citation | title=see OEIS A064984| url= https://oeis.org/A158982}}</ref> 

: <math> \Psi_{2^{k+1}}(2x) = 2T_{2^{k-1}}(x) .</math>

== Absolute value of the constant coefficient ==
The [[absolute value]] of the constant coefficient of <math>\Psi_n(x)</math> can be determined as follows:<ref>{{cite journal|author=C. Adiga, I. N. Cangul and H. N. Ramaswamy|year=2016|title=On the constant term of the minimal polynomial of <math>\cos\left(\frac{2\pi}{n}\right)</math> over <math>\mathbb Q</math>|journal=[[Filomat]]|volume=30|issue=4|pages=1097–1102|doi=10.2298/FIL1604097A|doi-access=free}}</ref>

: <math>|\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}</math>

== Generated algebraic number field ==
The [[algebraic number field]] <math>K_n = \mathbb{Q}\left(\zeta_n + \zeta_n^{-1}\right)</math> is  the maximal real subfield of a [[cyclotomic field]] <math>\mathbb Q(\zeta_n)</math>. If <math>\mathcal O_{K_n}</math> denotes the [[ring of integers]] of <math>K_n</math>, then <math>\mathcal O_{K_n} = \mathbb Z\left[\zeta_n + \zeta_n^{-1}\right]</math>. In other words, the [[set (mathematics)|set]] <math>\left\{1, \zeta_n + \zeta_n^{-1}, \ldots, \left(\zeta_n + \zeta_n^{-1}\right)^{\frac{\varphi(n)}{2} - 1}\right\}</math> is an integral basis of <math>\mathcal O_{K_n}</math>. In view of this, the [[Discriminant of an algebraic number field|discriminant of the algebraic number field]] <math>K_n</math> is equal to the [[Discriminant|discriminant of the polynomial]] <math>\Psi_n(x)</math>, that is<ref>{{cite journal|author=J. J. Liang|year=1976|title=On the integral basis of the maximal real subfield of a cyclotomic field|journal=[[Journal für die reine und angewandte Mathematik]]|volume=286-287|pages=223–226}}</ref>
: <math>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}</math>

== References ==
{{reflist}}

[[Category:Number theory]]
[[Category:Polynomials]]
[[Category:Trigonometry]]