Minimal Polynomials of Algebraic Cosine Values at Rational Multiples of π

Pinthira Tangsupphathawat Department of Mathematics

Phranakhon Rajabhat University

Bangkok 10220 Thailand

 t.pinthira@hotmail.com

Vichian Laohakosol Department of Mathematics

Faculty of Science Kasetsart University Bangkok 10900 Thailand

    fscivil@ku.ac.th

Abstract

Lehmer proved that the values of the cosine function evaluated at rational multiples of π are algebraic numbers. We show how to determine explicit, closed form expressions for the minimal polynomials of these algebraic numbers.

1 Introduction

The following information concerning algebraic values of the cosine (and certain trigonometric) functions evaluated at rational multiples of π is well-known [2], [5, Chapter 3].

Let n ∈ ℕ, k ∈ {1, 2, …, n} with n > 2 and gcd(k, n) = 1. Then the value 2 cos(2kπ/n) is an algebraic integer of degree φ(n)/2 whose minimal polynomial is ψ_{n}(x) ∈ ℤ[x], where

ψ_{n}(x + x^{−1}) = x^{−φ(n)/2}Φ_{n}(x),  (1)

with Φ_{n} being the nth cyclotomic polynomial and φ denoting the Euler’s totient function.

A simple proof of a special case of pure quadratic cosine values \sqrt{r} is given in [10] and extended in [8], which also contains details of algebraic cosine values of small degrees.

It is natural to ask whether there is an explicit, closed form expression for ψ_{n}(x). Since there is no explicit closed form expression for a general cyclotomic polynomial, Φ_{n}(x), such a task seems non-trivial. Recall that in order to find an expression for Φ_{n}(x), a usual approach is to use certain reduction formulas to write it as an algebraic expression involving known cyclotomic polynomials.

Our aim is to derive some reduction formulae, as with the case of cyclotomic polynomials, which will enable us to completely determine explicit forms of the polynomials ψ_{n}(x). In addition, we mention two remarks concerning the values of the cosine functions evaluated at an algebraic irrational multiple of π and certain constructible values.

2 Minimal polynomials

Since cos(2kπ/n) = cos(2π(n − k)/n), as k runs those integers in {1, 2, …, n} which are relatively prime to n, there are precisely φ(n)/2 (= deg ψ_{n}(x)) distinct values of cos(2kπ/n) which shows that these cosine values are all the roots of ψ_{n}(x) [9, Lemma, p. 473]. We base our investigation on the following explicit form of ψ_{p}(x), p odd prime, due to Surowski and McCombs [7].

Proposition 1. [7, Theorem 2.1] Let p = 2s + 1 be an odd prime. If ψ_{p}(x) ∈ ℤ[x] is the minimal polynomial of 2 cos(2π/p), then

ψ_{p}(x) = ∑_{j=0}^{s}(−1)^{j}σ_{j}x^{s−j},

where

σ_{2k} = (−1)^{k}\binom{s − k}{k}  (k = 0, 1, …, ⌊s/2⌋)
σ_{2k−1} = (−1)^{k}\binom{s − k}{k − 1}  (k = 1, …, ⌊(s + 1)/2⌋).

Incorporating this result with our preceding remark, we get Theorem 2. Let p = 2s + 1 be an odd prime. The minimal polynomial of 2 cos(2kπ/p), where k ∈ {1, 2, …, p}, gcd(k, p) = 1, is

ψ_{p}(x) = ∑_{j=0}^{⌊s/2⌋}(−1)^{j}\binom{s − j}{j}x^{s−2j} − ∑_{j=1}^{⌊(s+1)/2⌋}(−1)^{j}\binom{s − j}{j − 1}x^{s−(2j−1)}.

We also need some identities involving cyclotomic polynomials, [3, Chapter 2].

Lemma 3. Let q be a prime, and let m, e ∈ ℕ. Then

A. Φ_{q^{e}}(x) = Φ_{q} (x^{q^{e−1}}) = 1 + x^{q^{e−1}} + x^{2q^{e−1}} + ⋯ + x^{(q−1)q^{e−1}};

B. Φ_{mq^{e}}(x) = Φ_{mq} (x^{q^{e−1}});

C. Φ_{mq}(x) = \frac{Φ_{m}(x^{q})}{Φ_{m}(x)} provided that gcd(m, q) = 1.

In order to determine explicit forms of minimal polynomials for 2 cos(2kπ/n) for other positive integers n, Lehmer’s identity (1) indicates that we should first find explicit forms of x^{s} + x^{−s} as a polynomial in x + x^{−1}, which is done in the next lemma.

Lemma 4. For t ∈ ℕ, let X_{t} := x^{t} + x^{−t}, X := X_{1} = x + x^{−1}. Then

X_{2t} = X^{2t} − \{\binom{2t − 1}{1} + \binom{2t − 2}{0}\} X^{2t−2} + \{\binom{2t − 2}{2} + \binom{2t − 3}{1}\} X^{2t−4}

+ ⋯ + (−1)^{t−1} \{\binom{t + 1}{t − 1} + \binom{t}{t − 2}\} X^{2} + (−1)^{t} · 2

X_{2t+1} = X^{2t+1} − \{\binom{2t}{1} + \binom{2t − 1}{0}\} X^{2t−1} + \{\binom{2t − 1}{2} + \binom{2t − 2}{1}\} X^{2t−3}

+ ⋯ + (−1)^{t−1} \{\binom{t + 2}{t − 1} + \binom{t + 1}{t − 2}\} X^{3} + (−1)^{t} \{\binom{t + 1}{t} + \binom{t}{t − 1}\} X^{1},

i.e., in general, for s ∈ ℕ, we have

X_{s} = X^{s} − \{\binom{s − 1}{1} + \binom{s − 2}{0}\} X^{s−2} + \{\binom{s − 2}{2} + \binom{s − 3}{1}\} X^{s−4} + ⋯

+ (−1)^{⌊s/2⌋} \{\binom{s − ⌊s/2⌋}{⌊s/2⌋} + \binom{s − ⌊s/2⌋ − 1}{⌊s/2⌋ − 1}\} X^{s−2⌊s/2⌋}

= ∑_{k=0}^{⌊s/2⌋}(−1)^{k} \{\binom{s − k}{k} + \binom{s − k − 1}{k − 1}\} X^{s−2k},

with the convention that \binom{n}{r} = 0 for negative r.

Proof. Taking n = p = 2s + 1, an odd prime, Lehmer’s identity (1) becomes

x^{−s}Φ_{p}(x) = x^{− \frac{p−1}{2}} Φ_{p}(x) = ψ_{p} (x + x^{−1}) .  (2)

Equating the left hand side of (2) (using Lemma 3 A)

Φ_{p}(x) = x^{p−1} + x^{p−2} + ⋯ + x + 1 = x^{2s} + x^{2s−1} + ⋯ + x + 1,

with the right hand side (using Theorem 2), we get (x^{s} + x^{−s}) + (x^{s−1} + x^{−(s−1)}) + ⋯ + (x + x^{−1}) + 1

= \binom{s}{0} (x + x^{−1})^{s} − \binom{s − 1}{1} (x + x^{−1})^{s−2} + \binom{s − 2}{2} (x + x^{−1})^{s−4} − ⋯

+ (−1)^{⌊s/2⌋}\binom{s − ⌊s/2⌋}{⌊s/2⌋} (x + x^{−1})^{s−2⌊s/2⌋}

+ \binom{s − 1}{0} (x + x^{−1})^{s−1} − \binom{s − 2}{1} (x + x^{−1})^{s−3} + \binom{s − 3}{2} (x + x^{−1})^{s−5} − ⋯

+ (−1)^{⌊(s+1)/2+1⌋}\binom{s − ⌊(s + 1)/2⌋}{⌊(s + 1)/2⌋ − 1} (x + x^{−1})^{s−(2⌊(s+1)/2⌋−1)} .  (3)

Putting s = 2t, equating terms with even powers, we get X_{2t} + X_{2t−2} + ⋯ + X_{2} + 1 = (x^{2t} + x^{−2t}) + (x^{2t−2} + x^{−(2t−2)}) + ⋯ + (x^{2} + x^{−2}) + 1

= \binom{2t}{0} (x + x^{−1})^{2t} − \binom{2t − 1}{1} (x + x^{−1})^{2t−2} + \binom{2t − 2}{2} (x + x^{−1})^{2t−4} + ⋯

+ (−1)^{t−1}\binom{2t − t + 1}{t − 1} (x + x^{−1})^{2} + (−1)^{t}\binom{2t − t}{t}

= X^{2t} − \binom{2t − 1}{1}X^{2t−2} + \binom{2t − 2}{2}X^{2t−4} + ⋯ + (−1)^{t−1}\binom{t + 1}{t − 1}X^{2} + (−1)^{t}.  (4)

Replacing t by t − 1 in (4), we get X_{2t−2} + X_{2t−4} + ⋯ + X_{2} + 1

= X^{2t−2} − \binom{2t − 3}{1}X^{2t−4} + \binom{2t − 4}{2}X^{2t−6} + ⋯ + (−1)^{t−2}\binom{t}{t − 2}X^{2} + (−1)^{t−1}. (5)

Subtracting (5) from (4), we get the first assertion. The second assertion follows from equating terms with odd exponents in (3) and proceed similarly.

Lemmas 4 and 3 enable us to find an explicit form of any minimal polynomial through the following reduction identities.

Theorem 5. I. For odd prime p and e ∈ ℕ, the minimal polynomial of 2 cos(2kπ/p^{e}), where k ∈ {1, 2, …, p^{e}}, gcd(k, p) = 1, is

ψ_{p^{e}}(x) = ψ_{p} ( ∑_{k=0}^{⌊p^{e−1}/2⌋}(−1)^{k} \{\binom{p^{e−1} − k}{k} + \binom{p^{e−1} − k − 1}{k − 1}\} x^{p^{e−1}−2k}) .

II. If p is an odd prime, and e, m are positive integers with m ≥ 2, gcd(m, p) = 1, then the minimal polynomial of 2 cos(2kπ/mp^{e}), where k ∈ {1, 2, …, mp^{e}}, gcd(k, mp) = 1, satisfies

ψ_{mp^{e}}(x) = \frac{ψ_{m} (∑_{k=0}^{⌊p^{e}/2⌋}(−1)^{k} \{\binom{p^{e}−k}{k} + \binom{p^{e}−k−1}{k−1}\} x^{p^{e}−2k})}{ψ_{m} (∑_{k=0}^{⌊p^{e−1}/2⌋}(−1)^{k} \{\binom{p^{e−1}−k}{k} + \binom{p^{e−1}−k−1}{k−1}\} x^{p^{e−1}−2k})}.

III. The minimal polynomial of 2 cos(2π/2) is

ψ_{2}(x) = x + 2.

IV. For e ∈ ℕ, e ≥ 2, the minimal polynomial of 2 cos(2kπ/2^{e}), where k ∈ {1, 2, …, 2^{e}}, gcd(k, 2) = 1, is

ψ_{2^{e}}(x) = ∑_{k=0}^{2^{e−2}}(−1)^{k} \{\binom{2^{e−1} − k}{k} + \binom{2^{e−1} − k − 1}{k − 1}\} x^{2^{e−1}−2k} + 2.

Proof. I. Using Lehmer’s identity (1) and Lemma 3 A, we have

ψ_{p^{e}}(X) = ψ_{p^{e}} (x + x^{−1}) = x^{−φ(p^{e})/2}Φ_{p^{e}}(x) = x^{−p^{e−1}(p−1)/2}Φ_{p}(x^{p^{e−1}})

= ψ_{p} (x^{p^{e}−1} + x^{−p^{e−1}}) = ψ_{p}(X_{p^{e−1}})

= ψ_{p} ( ∑_{k=0}^{⌊p^{e−1}/2⌋}(−1)^{k} \{\binom{p^{e−1} − k}{k} + \binom{p^{e−1} − k − 1}{k − 1}\} X^{p^{e−1}−2k}) .

II. Using Lehmer’s identity (1), gcd(m, p) = 1, Lemma 3 B and C, we get

ψ_{mp^{e}}(X) = ψ_{mp^{e}} (x + x^{−1}) = x^{−φ(mp^{e})/2}Φ_{mp^{e}}(x) = x^{−φ(m)p^{e−1}(p−1)/2}Φ_{mp}(x^{p^{e−1}})

= \frac{(x^{p^{e}})^{−φ(m)/2} Φ_{m}(x^{p^{e}})}{(x^{p^{e−1}})^{−φ(m)/2} Φ_{m}(x^{p^{e−1}})} = \frac{ψ_{m}(x^{p^{e}} + x^{−p^{e}})}{ψ_{m}(x^{p^{e−1}} + x^{−p^{e−1}})} = \frac{ψ_{m}(X_{p^{e}})}{ψ_{m}(X_{p^{e−1}})}.

III. This is obvious from 2 cos(2π/2) = −2.

IV. Observe from Lehmer’s identity (1), Lemma 3 A, and Lemma 4 that

ψ_{2^{e}}(X) = ψ_{2^{e}} (x + x^{−1}) = x^{−φ(2^{e})/2}Φ_{2^{e}}(x) = x^{−2^{e−2}}Φ_{2}(x^{2^{e−1}}) = ψ_{2} (x^{2^{e−1}} + x^{−2^{e−1}})

= ψ_{2}(X_{2^{e−1}}) = ψ_{2} (∑_{k=0}^{2^{e−2}}(−1)^{k} \{\binom{2^{e−1} − k}{k} + \binom{2^{e−1} − k − 1}{k − 1}\} X^{2^{e−1}−2k}) ,

and the desired assertion follows from part III.

3 Two more remarks about values of the cosine function

We end this paper with two further related observations.

3.1 Values at irrational multiples of π

Since all the cosine values evaluated at rational multiples of π are algebraic, it is natural to ask about the values at irrational multiples of π. We give an answer to this question using the following result of Robinson [6, Theorem 3(a)], that is a nice consequence of the Lindeman-Weierstrass theorem and the Gelfond-Schneider theorem.

Proposition 6. If α, β are algebraic with αi (i = \sqrt{−1}) irrational, then cos(α log β), sin(α log β) and tan(α log β) for β ≠ 0, 1 are transcendental numbers regardless of the branch of the natural logarithm.

The answer to our question is given in the following theorem.

Theorem 7. If γ is an algebraic irrational number, then cos(γπ), sin(γπ), and tan(γπ) are transcendental numbers.

Proof. Taking β = exp(πi) = −1 and α as an algebraic number such that αi = γ is an algebraic irrational number, Proposition 6 shows that cos(γπ) = cos(αiπ) = cos(α log β) is transcendental, and similarly for the sine and tangent functions.

3.2 Constructible values

Since cos(2kπ/n) (n > 2, k ∈ {1, …, n}, gcd(k, n) = 1) is an algebraic number of degree φ(n)/2, another natural question is whether this algebraic number is constructible [1, Section 7.11]. The answer is an immediate consequence of the following result whose proof can be found in [4].

Theorem 8. Let k/n (n > 2) be a rational number with gcd(k, n) = 1. Then the algebraic integers 2 cos(2πk/n) are constructible if and only if φ(n) is a power of 2, i.e., if and only if n = 2^{α}p_{1}p_{2} ⋯ p_{r}, where α is a non-negative integer, p_{1}, …, p_{r} are distinct odd primes of the form 2^{β} + 1.

We are grateful to the referee for the following information. Theorem 8 is a well-known ancient result. Gauss proved in his Disquisitiones arithmeticae that the numbers 2 cos(2πk/n) are constructible (of course, it is enough to take k = 1); in particular, he stated the construction of the 17th-gon, that is a famous fact in the life of Gauss. The converse part is usually attributed to Wantzel (1838).

4 Acknowledgments

We thank the referee for several useful comments and information. The first author is supported by Phranakhon Rajabhat University. The second author is supported by the Faculty of Science, Kasetsart University, through the grant number RFG1-13.

References

[1] P. Grillet, Algebra, Wiley, 1999.

[2] D. H. Lehmer, A note on trigonometric algebraic numbers, Amer. Math. Monthly 40 (1933), 165–166.

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[4] A. Mostowski, Un théorème sur les nombres cos 2πk/n, Colloq. Math. 1 (1948), 195–196.

[5] I. Niven, Irrational Numbers, Carus Monographs, Vol. 11, The Mathematical Association of America, 1956.

[6] M. L. Robinson, On certain transcendental numbers, Michigan Math. J. 31 (1984), 95–98.

[7] D. Surowski and P. McCombs, Homogeneous polynomials and the minimal polynomial of cos(2π/n), Missouri J. Math. Sci. 15 (2003), 4–14.

[8] P. Tangsupphathawat, Algebraic trigonometric values at rational multiples of π, Acta et Commentationes Universitatis Tartuensis de Mathematica 18 (2014), 9–18.

[9] W. Watkins and J. Zeitlin, The minimal polynomial of cos(2π/n), Amer. Math. Monthly 100 (1993), 471–474.

[10] J. L. Varona, Rational values of the arccosine function, Central European J. Math. 4 (2006), 319–322.

2010 Mathematics Classification: Primary 11R04; Secondary 33B10, 11J72. Keywords: cosine function, algebraic value, multiple of π.

Received February 8 2016; revised version received March 15 2016. Published in Journal of Integer Sequences, March 19 2016.

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