Math. J. Okayama Univ. 56 (2014), 27–33

INTERSECTIVE POLYNOMIALS WITHGALOIS GROUP D5
Melisa J. Lavallee, Blair K. Spearman and Qiduan Yang
Abstract. We give an infinite family of intersective polynomials withGalois group D5 , the dihedral group of order 10.

1. Introduction
A monic polynomial f (x) with integer coefficients is called intersective if
it has a root modulo m for all positive integers m. Equivalently f (x) has aroot in the field of p-adic numbers Qp for all primes p. We call f (x) nontrivially intersective if it is intersective but has no rational root. Henceforthin this paper our polynomials are nontrivially intersective. It is known thatf (x) cannot be irreducible over Q. In fact if f (x) is irreducible over Q, thenthere exist prime numbers p for which the congruence f (x) ≡ 0(mod p) cannot be solved in Z as shown by Brandl, Bubboloni and Hupp [3, Propostion
1.2]. The existence of intersective polynomials depends on properties of the
Galois group. Let L denote the splitting field of f (x) and Gal(f ) denote theGalois group of f (x). Behrend and Bilu [2] gave a simple condition to decide
whether or not a polynomial is intersective. Sonn [11] gave a condition forf (x) to be intersective as well as a method for constructing such polynomials. To apply this method it suffices to check that Gal(f ) is n-coverable forsome n > 0, that is Gal(f ) is the union of conjugates of n proper subgroups,
the intersection of all the conjugates is trivial, and every decompositiongroup G(p) for p a prime ideal in L is contained in a conjugate of one ofthese proper subgroups. We recall that the decomposition group G(p) is theset of elements σ ∈ Gal(f ) such that σ(p) = p. This condition clearly holds
if Gal(f ) is n-coverable and every decomposition group is cyclic. Having
checked this condition, it remains to determine a set of n monic polynomials with integer coefficients, defining those subfields of L corresponding
to the n chosen proper subgroups. The product of these polynomials will
be intersective. Examples of intersective polynomials in general seem to bescarce. A single example of an intersective polynomial with Galois group
D5 the dihedral group of order 10 is given in Sonn [12]. The notation D10
is also used for this group. The Galois group D5 has the added interest
that it is 2-coverable so that the constructed intersective polynomial has
Mathematics Subject Classification. Primary 11R32; Secondary 11R09, 11Y40.
Key words and phrases. Intersective polynomial, Galois group, dihedal group,monogenic.
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two irreducible factors. In this paper we give an infinite parametric familyof intersective polynomials with Galois group D5 . We make use of a familyof polynomials studied by Lavallee, Spearman Williams and Yang in [7]. In
that paper it was shown that they define monogenic quintic fields, that isalgebraic number fields whose rings of integers have a power basis. While
the property of monogeneity is not necessary when constructing intersective
polynomials, it is helpful when dealing with parametric families. This isdue to the fact that we can employ a well known theorem of Dedekind on
ideal factorization to assist with the study of the decomposition groups. InSection 2, we recall some known properties of the polynomials we study andsome facts about ideal factorization in number fields, including the theorem of Dedekind. In Section 3, we prove our theorem, give an estimate fordensity of the values of b for which db is squarefree and finish with someexamples of intersective polynomials using our theorem. In Section 4 weconsider some related examples. We state our main theorem next.
Theorem 1.1. There exist infinitely many integers b such that
db := −4b3 − 28b2 − 24b − 47(1)

is squarefree. Let b be such an integer. If we setfb (x) = x5 − 2x4 + (b + 2)x3 − (2b + 1)x2 + bx + 1
then the polynomial
gb (x) = fb (x)(x2 − db )is intersective and has Galois group D5 . Moreover for infinitely many of theintegers b for which db is squarefree, the splitting fields of gb (x) are distinct.
2. The Parametric Family
The parametric family of polynomialsfb (x) = x5 − 2x4 + (b + 2)x3 − (2b + 1)x2 + bx + 1
for b an integer was studied in [7]. We record some of the properties of thesepolynomials and their related number fields, referring the reader to [7] fordetails. The polynomial fb (x) is irreducible over Q for all integers b, and ifdb , given by (1) is squarefree, then the Galois group of fb (x) is isomorphic toD5 . Now assume that√ db is squarefree. The quadratic subfield of the splitting
field of fb (x) is Q db with discriminant db , and if θ is a root of fb (x) thenQ (θ) has field discriminant d2b , and is monogenic with ring of integers Z [θ] .
Moreover infinitely many of the fields Q (θ) are distinct.
The proof of our theorem requires the following propositions concerning
ideal factorization in algebraic number fields. For the first proposition, FINTERSECTIVE D5 POLYNOMIALS

29and K denote algebraic number fields and DK denotes the ring of integersof K. We only state that part of this proposition which we will use.
Proposition 1. [8, Cor. 4.100] Let K/F be a Galois extension of number
fields with p a prime DK -ideal. If p is unramified in K/F then the decomposition group of p is cyclic.
The next proposition concerns the factorization of primes in a compositeof two extensions of the same field. For notation, K, K1 and K2 denote
algebraic number fields and R denotes the ring of integers of K.
Proposition 2. [9, p. 159] If p is a prime ideal of R unramified in both K1 /K
and K2 /K then it is also unramified in the composite extension K1 K2 /K.
For the next proposition we use the notation irrQ (θ) to denote the monic
minimal polynomial in Z [x] of the algebraic integer θ. In addition we usethe notation ind(θ) to denote the index of θ, as defined by the equationsD(θ)
ind(θ) =d(K)where D(θ) is the polynomial discriminant of irr(θ) and d(K) is the field
discriminant of K = Q (θ) .
Proposition 3. [1, Cor. 10.5.1] Let K = Q (θ) be an algebraic number fieldwith θ ∈ OK the ring of integers of K. Let p be a rational prime. Letf (x) = irrQ (θ) ∈ Z [x] .
Let − denote the natural map Z [x] → Zp [x] , where Zp = Z/pZ. Letf (x) = g1 (x)e1 · · · gr (x)er ,where g1 (x), . . . gr (x) are distinct monic irreducible polynomials in Zp [x] ,and e1 , . . . , er are positive integers. For i = 1, 2, . . . , r, let fi (x) be any monicpolynomial of Z [x] such that f i = gi and deg(fi ) = deg(gi ). Set
Pi = hp, fi (θ)i , i = 1, 2, . . . , r.
If ind(θ) 6≡ 0(mod p), then P1 , . . . , Pr are distinct prime ideals of OK with
hpi = P1e1 · · · Prer ,N (Pi ) = pdeg fi , i = 1, . . . , r.

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3. proof of theorem
Proof. The fact that there are infinitely many integers b such thatdb = −4b3 − 28b2 − 24b − 47
is squarefree follows from a theorem of Erdös [5]. It was noted in [12] that
D5 is 2-coverable with the subgroup of order 5 and a subgroup of order
2. In order to construct intersective polynomials, then as stated in theintroduction we first show that the decomposition group G(p) is cyclic foreach prime ideal p in the splitting field L of fb (x). If p is unramified in Lthen G(p) is cyclic by Proposition 1. Now let p be a ramified prime in Llying above the rational prime p. Since Gal(fb ) ≃ D5 , G(p) is cyclic if andonly if G(p) is a proper subgroup of D5 . As Gal(fb ) acts transitively on
the set of prime ideals in L lying above p we deduce that G(p) is a proper
subgroup of D5 if there are at least two prime ideals in L lying above√p. Ifθ denotes a root of fb (x), then L is the compositum of Q (θ) and Q dbso that p must ramify in at least one of these fields by Proposition 2. We
recall from Section 2 that these fields have discriminants db and d2b which
obviously contain the same prime factors. Therefore p must ramify in bothof these fields and in particular in Q (θ). If only one prime ideal p in L
lies above p it follows that only one prime ideal of Q (θ) lies above p. Since
Q (θ) is monogenic with ring of integers Z [θ] we have ind(θ) 6≡ 0(mod p).
Therefore we can apply Proposition 3 and conclude that the only possible
factorization of fb (x) modulo p, consistent with the previous statements
about the factorization of a ramified ideal hpi leads to r = 1 and e = 5 sothat(2)fb (x) ≡ (x + t)5 (mod p),

where t is an integer. Equating each coefficient of fb (x) − (x + t)5 to zero
modulo p gives us a set of congruences from which we will derive a contradiction. The coefficient of x4 is−2 − 5t,
and the constant term is
1 − t5 ,each of which is divisible by p. From the identity(625t4 − 250t3 + 100t2 − 40t + 16)(−2 − 5t) − 55 (1 − t5 ) = −7 · 11 · 41,
it follows that the prime p must be one of 7, 11 or 41. Suppose that p = 7.
Since
−2 − 5t ≡ 0 (mod 7),

INTERSECTIVE D5 POLYNOMIALS

31

we have(3)t ≡ 1 (mod 7).

Since p is ramified we havedb ≡ 0 (mod 7),
so that(4)b ≡ 4 (mod 7).

However the coefficient of x3 in fb (x) − (x + t)5 isb − 10t2 + 2,

which is not divisible by 7 if (3) and (4) hold. If p = 11 we are led in thesame way to the congruence(t, b) ≡ (4, 4) or (4, 7) (mod 11),
The pair (4, 4) is omitted because db is not squarefree, while the remaining
pair (4, 7) yields a contradiction as in the case p = 7. For the case p = 41
we are led to the congruence(t, b) ≡ (16, 25) (mod 41)
and this possibility leads to the same type of contradiction as in the casep = 7. Hence (2) is impossible. Thus, we conclude that more than one prime
ideal in L lies over p, so that all of the decomposition groups are proper andhence cyclic. The method of construction for an intersective polynomial
when Gal(f ) ≃ D5√requires
us to form the product of fb (x) with a defining
polynomial for Q db , thusgb (x) = fb (x)(x2 − db )is intersective and has Galois group D5 . Finally, since infinitely many of thealgebraic number fields Q (θ) are distinct as noted in the introduction√  toSection 2, we have that infinitely many of the splitting fields Q θ, db of
gb (x) are distinct.
Remark. The frequency with which the quantity db is squarefree is clarified
by a theorem of Hooley on the squarefree values of cubic polynomials given
in [6]. This theorem implies that for a positive constant C we haveS(b, x) ∼ Cx
whereS(b, x) = # {b : |b| ≤ x and db is squarefree} .

Thus the set of integers b for which db is squarefree has positive density.

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We close this section with some examples of intersective polynomials obtained from our theorem.
Example 1. For the following values of b we construct intersective polynomials using our theorem. All of the roots in C of the first two intersectivepolynomials are real, while in the last two examples only one root is real.
The splitting fields of these polynomials are distinct.
bdb−9 817
−8 40102−47

Intersective Polynomial(x5 − 2x4 − 7x3 + 17x2 − 9x + 1)(x2 − 817)
(x5 − 2x4 − 6x3 + 15x2 − 8x + 1)(x2 − 401)(x5 − 2x4 + 2x3 − x2 + 1)(x2 + 47)−239 (x5 − 2x4 + 4x3 − 5x2 + 2x + 1)(x2 + 239).
4. Related Examples

For the purpose of comparison with the D5 polynomials that we made useof in this paper we consider dihedral quintic trinomials x5 + ax + b where aand b denote rational numbers. A parametrization of these polynomials isgiven by Roland, Yui and Zagier [10]. It was shown in [13] that the quintic
fields defined by these polynomials have a common index divisor of 2, sothat none of these fields are monogenic. Nevertheless some of them give riseto intersective polynomials.
Example 2. If f (x) = x5 + 11x + 44 then Gal(f ) ≃ D5 . It can be checkedthat all of the decomposition groups are cyclic. To deal with the ramified
primes and associated decomposition groups, algorithms for factoring ideals
are given Cohen [4]. The quadratic subfield of the splitting field of f (x)√ inis Q −2 and so the method used in [12] shows that(x5 + 11x + 44)(x2 + 2)

is intersective.
Example 3. If f (x) = x5 −√5x+ 12
 then Gal(f ) ≃ D5 . The quadratic subfieldof the splitting field is Q −10 . However there is a single ideal lying above
5 in the splitting field of f (x) and the decomposition group of this ideal is
all of D5 . Following the method in [12] yields the polynomial(x5 − 5x + 12)(x2 + 10),which is not intersective since it has no root modulo 25.

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33

References[1] Ş. Alaca and K. S. Williams, Introductory Algebraic Number Theory, Cambridge University Press, Cambridge, 2004.
[2] D. Behrend and Y. Bilu, Polynomials with roots modulo every integer, Proc. Amer.
Math. Soc. 124 (1996), no. 6, 1663-1671.
[3] R. Brandl, D. Bubboloni and I. Hupp, Polynomials with roots mod p for all primes p.
J. Group Theory 4 (2001), 233-239.
[4] H. Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag,(1993).
[5] P. Erdös, Arithmetic properties of polynomials, J. London Math. Soc. 28 (1953), 416425.
[6] C. Hooley, Applications of sieve methods to the theory of numbers. Cambridge Tracts inMathematics, No. 70. Cambridge University Press, Cambridge-New York-Melbourne,
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[8] R. A. Mollin, Algebraic Number Theory, Chapman and Hall, 1999.
[9] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, Springer, thirdedition, 1990.
[10] G. Roland, N. Yui and D. Zagier, A parametric family of quintic polynomials withGalois group D5. J. Number Theory 15(1982), 137–142.
[11] J. Sonn, Polynomials with roots in Qp for all p. Proc. Amer. Math Soc. 136 (2008),
no. 6, 1955-1960.
[12] J. Sonn, Two Remarks On The Inverse Galois Problem For Intersective Polynomials,J. Theor. Nombres Bordeaux, 21, (2009), no. 2, 437-439.
[13] B. K. Spearman, K. S. Williams and Q. Yang, On the Common Index Divisors of aDihedral Field of Prime Degree. Int. J. Math. Math. Sci. (2007), Article ID 89713, 8pages.
Melisa J. Lavallee
Department of Mathematics and Statistics
University of British Columbia Okanagan
Kelowna, BC, Canada, V1V 1V7.
e-mail address: Melisa Lavallee@hotmail.com
Blair K. Spearman
Department of Mathematics and Statistics
University of British Columbia Okanagan
Kelowna, BC, Canada, V1V 1V7.
e-mail address: blair.spearman@ubc.ca
Qiduan Yang
Department of Mathematics and Statistics
University of British Columbia Okanagan
Kelowna, BC, Canada, V1V 1V7.
e-mail address: qiduan.yang@ubc.ca(Received November 15, 2011 )(Revised April 10, 2012 )