International Journal of Algebra, Vol. 10, 2016, no. 4, 163 - 170
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ija.2016.6319
On Estimates for the Number of Irreducible
Cyclotomic Factors of a Polynomial in
an Algebraic Number Field II
Ali H. Hakami
Department of Mathematics, Faculty of Science, Jazan University,
P.O. Box 277, Jazan, Postal Code: 45142, Saudi Arabia,
c 2016 Ali H. Hakami. This article is distributed under the Creative ComCopyright mons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
Abstract
Let k be a number field and F a polynomial in k[x] with degree ∂(F )
and F (0) 6= 0. As in our work [2] we consider the problem of estimating
the number of irreducible factors of F in k[x] in terms of ∂(F )p
and of
the height of F. In this paper we shall give bounds on the size ∂(F )
provided the factors are counted without their multiplicities e(n, s).
Mathematics Subject Classification: Primary 11R09; Secondary 11C08,
12D05, 11G15, 11R60
Keywords: Polynomials; polynomials factorization; irreducibility of polynomials; algebraic numbers
1
Introduction
Let k be an algebraic number field and let F (x) be a polynomial in k[x]
of degree ∂(F ) with F (0) 6= 0. As in [2] we shall consider the problem of
estimating the number of irreducible factors of F in k[x] p
in terms of ∂(F ) and
of the height of F. We shall give bounds on the size of ∂(F ) provided the
factors are counted without their multiplicities e(n, s).
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Ali H. Hakami
Let Φn (x) in Z[x] denote the n-th cyclotomic polynomial. We assume that
Φn factors in k[x] as
δ(k;n)
Y
Φn,s (x),
(1.1)
Φn (x) =
s=1
where each factor Φn,s is monic and irreducible in k[x]. If ζn is a primitive n-th
root of unity then each factor Φn,s has degree [k(ζn ) : k]. As Q(ζn )/Q is Galois
we have
ϕ(n)
.
(1.2)
[k(ζn ) : k] = [Q(ζn ) : k ∩ Q(ζn )] =
[k ∩ Q(ζn ) : Q]
It follows that
δ(k ; n) = [k ∩ Q(ζn ) : Q] 6 [k0 : Q],
(1.3)
where k0 ⊆ k is the maximum abelian subfield of k. Indeed, δ(k0 ; n) = δ(k ; n)
since each factor Φn,s (x) occurs in k0 [x]. Next we suppose that F (x) factors
into irreducible polynomials in k[x] as

(
)
∞ δ(k;n)
I
Y
 Y
Y
F (x) =
Φn,s (x)e(n,s)
fi (x)m(i) .
(1.4)
n=1 s=1

i=1
Here e(n, s) > 0, m(i) > 1, and fi (x), i = 1, 2, . . . , I, are distinct, irreducible,
noncyclotomic polynomials in k[x]. There will be no loss of generality if we
assume that F is monic and hence that each fi is monic and has ∂(fi ) > 1. Of
course e(n, s) = 0 for all but finitely many pairs {n, s}, has 1 6 s 6 δ(k; n).
Thus
total number of cyclotmic factors of F counted with multiplicity is
Pδ(k;n)
P∞ the
and the total number of noncyclotomic factors counted
s=1 e(n, s),P
n=1
with multiplicity is Ii=1 m(i), (see for example [5]). Obviously we have the
trivial bound
∞ δ(k;n)
I
X
X
X
e(n, s) +
m(i) 6 ∂(F ),
(1.5)
n=1 s=1
i=1
and in general nothing more can be said. However, if ∂(F ) is large compared
with log H(F ) we may expect to obtain sharper bounds.
Throughout this work, as in [2,3,4,5], c(k) is defined by




∞
Y
 X δ(k; l)2 ϕ(J/l)
X
1
−1
1+
c(k) = lim X
δ(k; n) =
,
X→∞


p(p
−
1)
ϕ(l)(J/l)


p
n=1
l|J
p-J
[k(ζn ):k]6X
(1.6)
where J = J(k) = min{j > 1 : k0 ⊆ Q(ζj )}. Of course the integer J is finite
by the theorem of Kronecker-Weber. In particular we have
c(Q) =
ζ(2)ζ(3)
,
ζ(6)
(1.7)
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On estimates for the number of irreducible cyclotomic factors ...
and more generally there is the upper bound
c(k) 6
ζ(2)ζ(3)
min [k0 : Q]2 , τ (J)[k0 : Q], J ,
ζ(6)
(1.8)
where τ is the divisor function.
2
Main Results
In ourpprevious work [2] one can see that all bounds in are never any smaller
than ∂(F ) log ∂(F ). However if these same factors
are simply counted withp
out their multiplicities e(n, s), bounds of size ∂(F ) are fairly obtained. Our
main result contains several results of this general form
Theorem 2.1 Let F (x) be a polynomial in k[x] factoring as indicated in
(1.4). Counting without multiplicity the cyclotomic factors satisfy the following
elementary bounds. Then for any ε > 0
∞ δ(k;n)
X
X
n=1
p
1 6 (1 + ε) 2c(k)∂(F ),
(2.1)
s=1
e(n,s)6=0
∞
X
p
1 6 (1 + ε) 2c(k)∂(F ),
(2.2)
n=1
e(n,s)6=0 for some s
∞
X
n=1
δ(k;n)
X
p
1
1 6 (1 + ε) 2c(k)∂(F ),
δ(k; n) s=1
(2.3)
e(n,s)6=0
whenever ∂(F ) > N0 (ε, k) is sufficiently large.
3
Auxiliary Lemmas
To prove the main results of this paper we need the following two lemmas
Lemma 3.1 ([3, Lemma 12]) The limits
1
c(k) = lim
N →∞ N
∞
X
n=1
[k(ζn ):k]6N
δ(k; n),
1
b(k) = lim
N →∞ N
∞
X
n=1
[k(ζn ):k]6N
1
(3.1)
166
Ali H. Hakami
exist, with
c(k) =
Y
1+
1
p(p−1)
∞
X
p-J
δ(k;m)2 ϕ(J/m)
,
ϕ(m)(J/m)
b(k) =
Y
1+
1
p(p−1)
∞
X
p-J
m|J
δ(k;m)2 ϕ(J/m)
,
ϕ(m)(J/m)
m|J
(3.2)
where J is the field constant
J = J(k) = min{j : k0 ⊆ Q(ζj )}.
(3.3)
These quantities possess the following upper bounds
c(k) 6
ζ(2)ζ(3)
ζ(6)
ζ(2)ζ(3)
min{d(J), [k0 : Q]}.
ζ(6)
(3.4)
min{[k0 : Q]2 , d(J)[k0 : Q], J}, b(k) 6
In the particular case k = Q
Y
c(k) = b(k) =
1+
p
1
p(p − 1)
=
ζ(2)ζ(3)
,
ζ(6)
and for a general cyclotomic field k = Q(ζJ )
−1 ζ(2)ζ(3) Y
1
1
c(Q(ζJ )) =
1+
1 − 2 J.
ζ(6)
p(p − 1)
p
(3.5)
(3.6)
p|J
Applying Lemma 3.1 we prove
Lemma 3.2
∞
X
ϕ(n) =
n=1
[k(ζn ):k]6N
∞
X
1
(1 + o(1)) c(k)N 2 ,
2
[k(ζn ) : k] =
n=1
[k(ζn ):k]6N
1
(1 + o(1)) b(k)N 2 .
2
(3.7)
(3.8)
Proof. The above lemma can be easily obtained from the definition of c(k)
and b(k) via partial summation. Let
b(m, k) =
∞
X
δ(k; n),
(3.9)
n=1
[k(ζn ):k]
then
∞
X
n=1
[k(ζn ):k]
ϕ(n) =
∞
X
n=1
[k(ζn ):k]
δ(k; n)[k(ζn ) : k] =
X
m6N
mb(m;k),
(3.10)
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On estimates for the number of irreducible cyclotomic factors ...
where recall by definition
X
∞
X
b(m, k) =
m6N
δ(k; n) = (1 + o(1))c(k)N.
(3.11)
n=1
[k(ζn ):k]6N
So applying partial summation (e.g. Hardy and Wright [1, Theorem 421])
∞
X
Z
ϕ(n) = N (1 + o(1)c(k)N −
N
N (1 + o(1))c(k)xdx
1
n=1
[k(ζn ):k]
1
= c(k)N 2 (1 + o(1)).
2
(3.12)
The sum (3.8) follows in the same way.
4
Proof of Theorem 2.1
First, arrange the cyclotomic polynomials Φn,s (x) in k[x] in order of increasing
degree. Let Pm (x) denote the m-th polynomial and Qm (x) the m-th Φn,1 (x) to
appear in the list. Similarly order the positive integers n in terms of increasing
ϕ(n) with nm denoting the m-th element of this list. Note, when k = Q we
have Pm (x) = Qm (x) = Φnm (x).
From the definition of b(k) and c(k) it is easy to see that once ∂(Qm ) and
∂(Pm ) are > N0 (ε, k)
m6
∞
X
δ(k;n)
X
1 6 (1 + ε)c(k)∂(Pm ).
(4.1)
1 6 (1 + ε)b(k)∂(Qm ).
(4.2)
n=1
s=1
[k(ζn ):k]6∂(Pm )
m6
∞
X
δ(k;n)
X
n=1
s=1
[k(ζn ):k]6∂(Pm )
Consequently once m > m0 (ε, k)
∂(Pm ) > (1 − ε)
∂(Qm ) > (1 − ε)
m
c(k)
m
b(k)
,
(4.3)
.
(4.4)
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Ali H. Hakami
In particular from the case k = Q
ϕ(nm ) > (1 − ε)
m
c(Q)
,
(4.5)
for all m > m0 (ε) sufficiently large.
Let F (x) be our polynomial factoring in k[x] as shown in (1.4) and use
U, V and W to denote the required sums;
U=
∞ δ(k;n)
X
X
1,
(4.6)
n=1 s=1
∞
X
V =
1,
(4.7)
e(n,s)6=0 for some s
W =
∞
X
n=1
δ(k;n)
X
1
1.
δ(k; n) s=1
(4.8)
Then, by (3.7) and (4.3), as long as U > U0 (ε, k)
∂(F ) >
∞ δ(k;n)
X
X
n=1
>
∂(Φn,s ) >
s=1
e(n,s)6=0
∞
X
U
X
∂(Pi )
i=1
δ(k;n)
X
n=1
s=1
[k(ζn ):k]<∂(PU )
∂(Φn,s ) =
∞
X
ϕ(n)
n=1
[k(ζn ):k]<∂(PU )
2 1
ε
1
U
2
1−
c(k)∂(PU ) > (1 − ε)
>
.
2
2
2
c(k)
(4.9)
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On estimates for the number of irreducible cyclotomic factors ...
Similarly, from (3.8) and (4.4) when V > V0 (ε, k)
∞
X
∂(F ) >
∂(Φn,s ) >
V
X
n=1
e(n,s)6=0 for some s
∂(Qi )
i=1
!
>
X
ϕ(ni ) + ϕ(mW )
i6W
=
X
X
h(mi )−
i>W
X
(1 − h(mi ))
i6W
ϕ(ni ) + ϕ(mW )(W − [W ])
i6W
>
∞
X
ϕ(n) >
n=1
ϕ(n)<ϕ(nW )
1
> (1 − ε)
2
W2
c(Q)
ε
1
1−
c(Q)ϕ(nW )2
2
2
.
(4.10)
Hence once ∂(F ) > N0 (ε) we obtain the desired inequality
p
W 6 (1 + ε) 2c(Q)∂(F ).
(4.11)
The examples
GN,k (x) =
HN,k (x) =
LN (x) =
∞
Y
n=1
[k(ζn ):k]6N
∞
Y
Φn (x),
(4.12)
Φn,1 (x),
(4.13)
n=1
[k(ζn ):k]6N
∞
Y
Φn (x),
(4.14)
n=1
ϕ(n)6N
show respectively the sharpness of (2.1), (2.2) and (2.3).
Acknowledgments. The author is grateful to Jazan University (JU) for
supporting this research. He would also like to thank Professor Matthew Johnson (TU) who read carefully the earlier versions of this paper and suggested
some improvements.
170
Ali H. Hakami
References
[1] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers,
Fifth Edition, Cambridge Univ. Press, 1979.
[2] A. Hakami, On estimates for the number of irreducible cyclotomic factors
of a polynomial in an algebraic number field II, to be appear in Adv. Appl.
Math. Sci., (2016).
[3] C. G. Pinner and J. D. Vaaler, The number of irreducible factors of a
polynomial (I), Trans. Amer. Math. Soc., 339 (1993), 809-834.
http://dx.doi.org/10.1090/s0002-9947-1993-1150018-x
[4] C. G. Pinner and J. D. Vaaler, The number of irreducible factors of a
polynomial (II), Acta Arithmetica, LXXVIII.2 (1996), 125-142.
[5] C. G. Pinner and J. D. Vaaler, The number of irreducible factors of a polynomial (III), Chapter in Number Theory in Progress, Vol. 1 (ZakopaneKoscielisko, 1997), 395-405, de Gruyter, Berlin, 1999.
Received: March 21, 2016; Published: May 5, 2016