Some Properties of the Minimal Polynomials of 2cos(π /q) for
odd q
Birsen Ozgur∗ , Musa Demirci† , Aysun Yurttas∗∗ and I. Naci Cangul‡
∗
Uludag University, Faculty of Science, Department of Mathematics, Bursa-Turkey, bı[email protected]
Uludag University, Faculty of Science, Department of Mathematics, Bursa-Turkey, [email protected]
∗∗
Uludag University, Faculty of Science, Department of Mathematics, Bursa-Turkey, [email protected]
‡
Uludag University, Faculty of Science, Department of Mathematics, Bursa-Turkey, [email protected]
†
Abstract. The number λq = 2 cos π /q, q ∈ N, q ≥ 3, appears in the study of Hecke groups which are Fuchsian groups, and
in the study of regular polyhedra. There are many results about the minimal polynomial of this algebraic number. Here we
obtain the minimal polynomial of this number by means of the better known Chebycheff polynomials for odd q and give some
of their properties.
Keywords: Hecke groups, roots of unity, minimal polynomials, Chebycheff polynomials.
PACS: 2010 MSC: 05E35, 33C45, 33C50, 33D45.
INTRODUCTION
For n ∈ N, the n-th Chebycheff polynomial Tn (x) is defined by
Tn (x) = cos(n cos−1 x),
where x ∈ R, |x| ≤ 1, or Tn (cos θ ) = cos(nθ ), θ ∈ R. We get a normalisation of Tn given by Dn (x) = 2Tn (x/2) =
2 cos(n arccos(x/2)) = 2 cos nθ where x = D1 (x) = 2 cos θ ; x, θ ∈ R, |x| ≤ 2, n ∈ N.
The minimal polynomial of λq = 2 cos(π /q) for odd q is given in [1, 3]. In this work we search for the minimal
polynomial of this algebraic number λq when q is even and find this polynomial in terms of normalized Chebycheff
polynomials. The number λq plays an important role in the theory of Hecke groups, which are the discrete subgroups
of PSL(2, R) generated by two linear fractional transformations
R(z) = −1/z
T (z) = z + λq ,
and
see [1]. λq is also used in the geometry of polyhedra.
The polynomials Dn are explicitly given in [4] as
[n/2]
n
Dn (x) = ∑ (−1)
n
−
i
i=0
i
n−i
i
xn−2i
where [α ] denotes the greatest integer less than or equal to α . Therefore deg(Dn (x)) = n.
The first few Dn ’s are D1 (x) = x, D2 (x) = x2 − 2, D3 (x) = x3 − 3x, D4 (x) = x4 − 4x2 + 2, D5 (x) = x5 − 5x3 + 5x. For
convenience we define D0 (x) = 2.
In [1], the fourth author obtained the minimal polynomial Pq∗ of λq by means of the Tn ’s and Dn ’s. It was also proven
that the degree of Pq∗ (x) is ϕ (2q)/2, where ϕ (q) denotes the Euler function. Here we give a relatively simpler formula
for Pq∗ .
Let ζ = eiπ /q so that λq = ζ + ζ −1 . Then ζ q + 1 = 0. As q is odd, (ζ + 1)(ζ q−1 − ζ q−2 + ... − ζ + 1) = 0, i. e.
q−1
ζ
− ζ q−2 + ... − ζ + 1 = 0. Hence
ζ (q−1) / 2 (ζ (q−1) / 2 − ζ (q−3) / 2 + ... − ζ (3−q) / 2 + ζ (1−q) / 2 ) = 0,
i.e.
(ζ (q−1) / 2 + ζ (1−q) / 2 ) − (ζ (q−3) / 2 + ζ (3−q) / 2 ) + ... − ε (ζ + ζ −1 ) + ε = 0
Numerical Analysis and Applied Mathematics ICNAAM 2011
AIP Conf. Proc. 1389, 353-356 (2011); doi: 10.1063/1.3636737
© 2011 American Institute of Physics 978-0-7354-0956-9/$30.00
353
where
ε=
(q−1)/2
The last equation is equivalent to ∑k=0
+1 i f q ≡ 1 mod 4
.
−1 i f q ≡ −1 mod 4
(−1)k D q−1 −k = 0. From now on we denote the polynomial on the left hand
2
side by Pq (x). We have deg Pq (x) = q−1
2 .
We denote by Wq (Wq∗ respectively) the set of roots of Pq (Pq∗ respectively) in the simple extension Q(λq ). Then
Theorem 1 Wq = {2 cos π /q, 2 cos 3π /q, ..., 2 cos(q − 2)π /q}.
Proof ζ q = −1 = eiπ , so ζk = ei(π /q+2kπ /q) , k = 1, 2, ..., q − 1. But eiπ /q .ei(2q−1)π /q = 1, ei3π /q .ei(2q−3)π /q =
1, ..., ei(q−2)π /q .ei(q+2)π /q = 1 and eiπ /q + e−iπ /q = λq , ei3π /q + e−i3π /q = 2 cos 3π /q, ..., ei(q−2)π /q + ei(q+2)π /q =
2 cos(q − 2)π /q. Therefore
Pq (x) = (x − 2 cos π /q)(x − 2 cos 3π /q)...(x − 2 cos(q − 2)π /q).
Theorem 2 If p | q then Wp ⊂ Wq .
Proof Let q = p.k, k ∈ N. Then
Wp = {2 cos kπ /q, 2 cos 3kπ /q, ..., 2 cos(q − 2)kπ /q }
and
Wq = {2 cos π /q, 2 cos 3π /q, ..., 2 cos(q − 2)π /q }.
We want to show that the first and the last elements in Wp are in Wq implying the result as the coefficients of the
angles π /q and kπ /q are successive odd numbers. As k is odd and k | q, 2 cos kπ /q ∈ Wq . Consider 2 cos(p − 2)kπ /q.
As (p − 2)k < p.k = q and (p − 2)k is odd, this is also in Wq .
We now built the minimal polynomial Pq∗ successively, beginning with prime q.
Theorem 3 Let q = p be an odd prime. Then Pp is irreducible and Pp∗ = Pp .
p−1
∗
Proof By (1), deg Pp (x) = p−1
2 . Also ϕ (2p)/2 = 2 . Now Pq and Pq are both polynomials of λq , the latter being
∗
minimal. In other words we obtain Pq by dividing Pq by several factors. As they both have the same degree, they must
be equal.
Theorem 4 Let q = p1 .p2 with p1 , p2 distinct odd primes. Then Pq is reducible and
Pq∗ =
Pq
∗
Pp1 .Pp∗2
=
Pq
.
Pp1 .Pp2
Proof By Theorem 2, Wpi ⊂ Wq for i = 1, 2, i.e. both Pp1 and Pp2 divide Pq . As polynomials form an Eulerian ring we
have
/
Lemma 5 (p1 , p2 ) = 1 iff Wp1 ∩Wp2 = 0.
354
By Lemma 5 we guarantee that Wp1 and Wp2 have no common element implying that no linear factor divides Pq
twice when it is divided by Pp1 and Pp2 together. This is necessary, and in fact very important, as all roots of Pq are
distinct.
Let Pq denote the polynomial obtained from Pq by dividing it by Pp1 and Pp2 . It has degree
p1 − 1 p2 − 1
q−1
−
+
deg Pq (x) =
2
2
2
(p1 − 1)(p2 − 1)
=
2
which is equal to the deg Pq∗ (x) of the minimal polynomial Pq∗ . Hence the result follows. Let us see this with an example.
Theorem 6 Let q = p2 , p odd prime. Then Pq is reducible and Pq∗ =
Pq
Pp .
Proof It is similar to the proof of Theorem 3.
Note that Theorems 4 and 1 suggest us to divide Pq by Pd for each divisor d, to obtain the minimal polynomial Pq∗ .
This process is necessary but not always enough as we shall see. In these theorems, q = p1 .p2 and q = p2 and therefore
all proper divisors of q are prime. In fact these two are the only values of q having this property. For all other q, there
are some non-prime divisors of q. Some of those will not be comprime. If d1 and d2 are two such divisors with (d1 ,
d2 ) = d > 1, then Pd1 and Pd2 are not comprime either. Similarly to Lemma 5, we have
Lemma 7 Let q be odd. Let d1 , d2 | q with (d1 , d2 ) = d > 1. Then
/
Wd1 ∩Wd2 = 0.
In fact
Lemma 8 Let q be odd. Let d1 , d2 | q. Then (d1 , d2 ) = d > 1 iff
/
Wd1 ∩Wd2 = Wd = 0.
Proof As d | di for i = 1, 2, by Theorem 2, Wd ⊂ Wdi and therefore
Wd ⊂ Wd1 ∩Wd2 .
Let now α ∈ Wd1 ∩Wd2 . Then α ∈ Wd1 and α ∈ Wd2 and hence α is a root of Pd1 and Pd2 . Let α = 2 cos mπ /n, such that
m, n are comprime. Here m < n, necessarily. As α is a root of Pd1 and Pd2 , n | d1 and n | d2 . Therefore n | (d1 , d2 ) = d.
Let d = l.n, l ∈ N. Then 1n = dl and therefore α = 2 cos lmπ /d ∈ Wd , as l.m is odd and 1 < l.m ≤ d − 2. Indeed
l.m < l.n = d and therefore l.m ≤ d − 2. Hence we have Wd = Wd1 ∩Wd2 . As λd ∈ Wd1 ∩Wd2 , Wd = 0.
/
Recall that we divide Pq by Pdi , di | q, to find Pq∗ . When two such divisors are comprime then corresponding P
polynomials have common factors. Therefore Pq is divided twice by the same factor. But we cannot do this as all
factors of Pq are different. For this reason, to find the Pq∗ , we must multiply Pq by Pd for every two comprime divisors
di , d j such that (di , d j ) = d, before dividing Pq by Pdi and Pd j . We can see the application of this in the following
theorem:
Theorem 9 Let q = pn , p odd prime, n ≥ 2, n ∈ N. Then Pq is reducible and
Pq∗ =
Pq
.
Ppn−1
Proof For n = 2 we have the result by Theorem 6. Now for n = 3, the only proper divisors of q are p and p2 . To obtain
Pp∗3 , we first divide Pp3 by Pp and Pp2 . But as Wp ⊂ Wp2 , not to divide Pp3 twice by the same factors is only possible by
dividing Pp2 by Pp first and then doing as before. i.e.
Pp∗3 =
Pp3
P2
Pp . Ppp
355
=
Pp3
Pp2
as required. Similarly
Pp∗n =
Ppn
Pp2 Pp3
P n−1
Pp . Pp . P 2 .. ... . Ppn−2
p
p
=
Ppn
.
Ppn−1
Theorem 10 Let q be odd. Then
Pq∗ =
Pq
.
∏ Pd∗
d|q,d
=1,q
Proof It follows from Lemma 8 and the discussion afterwards.
Note that the number of divisors Pd∗ in the denominator could be made less using Theorems 1.4, 1.6 and 1.9. e.g. if
q = p21 .p2 then
Pq∗ =
Pq
∗
∗
Pp1 .Pp2 .Pp∗2 .Pp∗1 p2
1
Pq
=
Pp2
Pp1 p2
.P
1 p2
Pp1 .Pp2 . Pp1 . Pp
1
=
Pq .Pp1
.
Pp2 .Pp1 p2
1
Acknowledgements The fourth author is supported by the Commission of Scientific Research Projects of Uludag
University, Project numbers 2006/40, 2008/31 and 2008/54.
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123(1998), 59–74.
4. H. Weber. Traite d’algebre Superieure, I. Gauthier-Villars, Paris, 1898.
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