Minimal polynomial of 2cos(2pi/n) explained

In number theory, the real parts of the roots of unity are related to one-another by means of the minimal polynomial of

2 \cos( 2 \pi/n). 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 \cos\left( 2 k\pi/n\right) with k coprime with n.

Formal definition

n \geq 1 , the minimal polynomial \Psi _{n(x)} of 2 \cos( 2 \pi/n) is the non-zero monic polynomial of smallest degree for which \Psi _{n\left(2\cos(2\pi/n)\right)} = 0 .

For every, the polynomial

\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 2 \cos\left( 2 k\pi/n\right) with k coprime with n and either 1 \le k < n or k = n =1 . These roots are twice the real parts of the primitive th roots of unity. The number of integers k relatively prime to n is given by Euler's totient function \varphi(n); it follows that the degree of \Psi _{n(x)} is 1 for n =1 , 2 and \varphi(n)/ 2 for n\geq 3 .

The first two polynomials are

\Psi _{1(x)} = x - 2 and \Psi _{2(x)} = x + 2 .

The polynomials

\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

\Psi _{n(x)} are\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} - 2 x - 1 \\ \Psi _{8(x)} & = x ^{2} - 2 \\ \Psi _{9(x)} & = x ^{3} - 3 x + 1 \\ \Psi _{10} (x) & = x ^{2} - x - 1 \\ \Psi _{11} (x) & = x ^{5} + x ^{4} - 4 x ^{3} - 3 x ^{2} +3 x + 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}

Explicit form if n is odd

If

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

Putting

n =2 m +1 and\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} --++ … \\ & = \sum ^{m} _{k =0} ( -1 ) ^{\lfloor} \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} --++ … \\ & = \sum ^{m} _{k =0} ( -1 ) ^{\lfloor} \binom {\lfloor (m + k)/ 2 \rfloor}{k} x ^{k} , \end{align} then we have \Psi _{p} (x) = \chi _{p} (x) for primes p .

If

n is odd but not a prime, the same polynomial \chi _{n} (x) , as can be expected, is reducible and, corresponding to the structure of the cyclotomic polynomials \Phi _{d} (x) reflected by the formula \prod _{d\mid} \Phi _{d(x)} = x ^{n} - 1 , turns out to be just the product of all \Psi _{d} (x) for the divisors d> 1 of n , including n itself: \prod _{d\mid} \Psi _{d} ( x) = \chi _{n} ( x). This means that the \Psi _{d} (x) are exactly the irreducible factors of \chi _{n} (x) , which allows to easily obtain \Psi _{d} (x) for any odd d , knowing its degree (1)/(2) \varphi(d) . For example, \begin{align} \chi _{15} ( x)& = x ^{7} + x ^{6} - 6 x ^{5} - 5 x ^{4} + 10 x ^{3} + 6 x ^{2} - 4 x - 1 \\ & = (x + 1 )(x ^{2} + x - 1 )(x ^{4} - x ^{3} - 4 x ^{2} + 4 x + 1 )\\ & = \Psi _{3} (x) ⋅ \Psi _{5} (x) ⋅ \Psi _{15} (x). \end{align}

Explicit form if n is even

From the below formula in terms of Chebyshev polynomials and the product formula for odd

n above, we can derive for even n \prod _{d\mid} \Psi _{d} ( x) = ( \chi _{n+1} (x) + \chi _{n-1} (x) ) .

Independently of this, if

n =2 ^{k} is an even prime power, we have for k\ge 2 the recursion (see)\Psi _{2 ^{k+1}} ( x) = (\Psi _{2 ^{k}} ( x)) ^{2-2} ,starting with \Psi _{4} ( x) = x .

Roots

The roots of

\Psi _{n(x)} are given by 2 \cos\left( (2 \pi k)/(n) \right) ,^[1] where 1 \leq k < (n)/(2) and \gcd(k, n) = 1 . Since \Psi _{n(x)} is monic, we have\Psi _{n(x)} = \displaystyle\prod _{\begin{array} {c} 1 \leq k < (n)/(2) \ \gcd(k, n) = 1 \end{array} } \left(x -2 \cos\left( (2 \pi k)/(n) \right)\right). \cos(x) is even, we find that 2 \cos\left( (2 \pi k)/(n) \right) is an algebraic integer for any positive integer n and any integer k .

Relation to the cyclotomic polynomials

For a positive integer

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

Relation to Chebyshev polynomials

In 1993, Watkins and Zeitlin established the following relation between

\Psi _{n(x)} and Chebyshev polynomials of the first kind.

If

n = 2 s + 1 is odd, then\prod _{d} \Psi _{d(2x)} = 2 (T _{s} (x) - T _{s(x)),}

and if

n = 2 s is even, then\prod _{d} \Psi _{d(2x)} = 2 (T _{s} (x) - T _{s} (x)).

If

n is a power of 2 , we have moreover directly \Psi _{2 ^{k+1}} ( 2 x) = 2 T _{2 ^{k-1}} (x) .

Absolute value of the constant coefficient

The absolute value of the constant coefficient of

\Psi _{n(x)} can be determined as follows:^[3] |\Psi _{n(0)|} = \begin{cases} 0 & if n = 4 ,\\ 2 & if n = 2 ^{k,} k \geq 0 , k ≠ 2 ,\ p & if n = 4 p ^{k,} k \geq 1 , p > 2 prime, \\ 1 & otherwise. \end{cases}

Generated algebraic number field

K _{n} = Q \left(\zeta _{n} + \zeta ^{-1} _{n} \right) is the maximal real subfield of a cyclotomic field Q(\zeta _{n)} . If l O _{K _{n}} denotes the ring of integers of K _{n} , then l O _{K _{n}} = Z\left[\zeta _{n} + \zeta ^{-1} _{n} \right] . In other words, the set \left\{ 1 , \zeta _{n} + \zeta ^{-1} _{n} , \ldots, \left(\zeta _{n} + \zeta ^{-1} _{n} \right) ^{(\varphi(n))/(- 1 2)} \right\} is an integral basis of l O _{K _{n}} . In view of this, the discriminant of the algebraic number field K _{n} is equal to the discriminant of the polynomial \Psi _{n(x)} , that is^[4] D _{K _{n}} = \begin{cases} 2 ^{(m -1 ) 2 ^{m} -1} & if n = 2 ^{m,} m > 2 ,\\p ^{(mp ^{m} - (m +1 )p ^{m} -1 )/ 2} & if n = p ^{m or 2p} ^{m,} p > 2 prime ,\ \left(\prod ^{\omega(n)} _{i =1} p ^{e _{i} -1 /(p _{i} -1 )} _{i} \right) ^{(\varphi(n))/(2)} & if \omega(n) > 1 , k ≠ 2 p ^{m.\end{cases}}

Notes and References

1. W. Watkins and J. Zeitlin. 1993. The minimal polynomial of \cos( 2 \pi/n) . The American Mathematical Monthly. 100. 5. 471–474. 10.2307/2324301. 2324301. free.
2. D. H. Lehmer. 1933. A note on trigonometric algebraic numbers. The American Mathematical Monthly. 40. 3. 165–166. 10.2307/2301023. 2301023.
3. C. Adiga, I. N. Cangul and H. N. Ramaswamy. 2016. On the constant term of the minimal polynomial of \cos\left( (2 \pi)/(n) \right) over Q . Filomat. 30. 4. 1097–1102. 10.2298/FIL1604097A. free.
4. J. J. Liang. 1976. On the integral basis of the maximal real subfield of a cyclotomic field. Journal für die reine und angewandte Mathematik. 286-287. 223–226.