We give an infinite family of intersective polynomials with
Galois group D5, the dihedral group of order 10

Lavallee, Melisa J.

Spearman, Blair K.

Yang, Qiduan

English

Okayama University Digital Information Repository

INTERSECTIVE POLYNOMIALS WITH GALOIS GROUP D<sub>5</sub>

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Math. J. Okayama Univ. 56 (2014), 27–33INTERSECTIVE POLYNOMIALS WITHGALOIS GROUP D5Melisa J. Lavallee, Blair K. Spearman and Qiduan YangAbstract. We give an infinite family of intersective polynomials withGalois group D5, the dihedral group of order 10.1. IntroductionA monic polynomial f(x) with integer coefficients is called intersective ifit 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) nontriv-ially 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) can-not be solved in Z as shown by Brandl, Bubboloni and Hupp [3, Propostion1.2]. The existence of intersective polynomials depends on properties of theGalois 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 decidewhether or not a polynomial is intersective. Sonn [11] gave a condition forf(x) to be intersective as well as a method for constructing such polynomi-als. 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 holdsif Gal(f) is n-coverable and every decomposition group is cyclic. Havingchecked this condition, it remains to determine a set of n monic polyno-mials with integer coefficients, defining those subfields of L correspondingto the n chosen proper subgroups. The product of these polynomials willbe intersective. Examples of intersective polynomials in general seem to bescarce. A single example of an intersective polynomial with Galois groupD5 the dihedral group of order 10 is given in Sonn [12]. The notation D10is also used for this group. The Galois group D5 has the added interestthat it is 2-coverable so that the constructed intersective polynomial hasMathematics Subject Classification. Primary 11R32; Secondary 11R09, 11Y40.Key words and phrases. Intersective polynomial, Galois group, dihedal group,monogenic.2728 M. LAVALLEE, B. SPEARMAN AND Q. YANGtwo 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]. Inthat paper it was shown that they define monogenic quintic fields, that isalgebraic number fields whose rings of integers have a power basis. Whilethe property of monogeneity is not necessary when constructing intersectivepolynomials, it is helpful when dealing with parametric families. This isdue to the fact that we can employ a well known theorem of Dedekind onideal 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 theo-rem 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(1) db := −4b3 − 28b2 − 24b− 47is squarefree. Let b be such an integer. If we setfb(x) = x5 − 2x4 + (b+ 2)x3 − (2b+ 1)x2 + bx+ 1then the polynomialgb(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 FamilyThe parametric family of polynomialsfb(x) = x5 − 2x4 + (b+ 2)x3 − (2b+ 1)x2 + bx+ 1for 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 splittingfield 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 concerningideal factorization in algebraic number fields. For the first proposition, FINTERSECTIVE 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 numberfields with p a prime DK-ideal. If p is unramified in K/F then the decom-position 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 denotealgebraic 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/Kand K2/K then it is also unramified in the composite extension K1K2/K.For the next proposition we use the notation irrQ (θ) to denote the monicminimal 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 equationind(θ) =√D(θ)d(K)where D(θ) is the polynomial discriminant of irr(θ) and d(K) is the fielddiscriminant 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). SetPi = 〈p, fi(θ)〉 , i = 1, 2, . . . , r.If ind(θ) 6≡ 0(mod p), then P1, . . . , Pr are distinct prime ideals of OK with〈p〉 = P e11 · · ·P err ,N(Pi) = pdeg fi , i = 1, . . . , r.30 M. LAVALLEE, B. SPEARMAN AND Q. YANG3. proof of theoremProof. The fact that there are infinitely many integers b such thatdb = −4b3 − 28b2 − 24b− 47is squarefree follows from a theorem of Erdo¨s [5]. It was noted in [12] thatD5 is 2-coverable with the subgroup of order 5 and a subgroup of order2. 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 onthe set of prime ideals in L lying above p we deduce that G(p) is a propersubgroup 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(√db)so that p must ramify in at least one of these fields by Proposition 2. Werecall from Section 2 that these fields have discriminants db and d2b whichobviously 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 Llies above p it follows that only one prime ideal of Q (θ) lies above p. SinceQ (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 possiblefactorization of fb(x) modulo p, consistent with the previous statementsabout the factorization of a ramified ideal 〈p〉 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 zeromodulo p gives us a set of congruences from which we will derive a contra-diction. The coefficient of x4 is−2− 5t,and the constant term is1− 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 31we 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 remainingpair (4, 7) yields a contradiction as in the case p = 7. For the case p = 41we 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 primeideal in L lies over p, so that all of the decomposition groups are proper andhence cyclic. The method of construction for an intersective polynomialwhen Gal(f) ≃ D5 requires us to form the product of fb(x) with a definingpolynomial 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)ofgb(x) are distinct. Remark. The frequency with which the quantity db is squarefree is clarifiedby a theorem of Hooley on the squarefree values of cubic polynomials givenin [6]. This theorem implies that for a positive constant C we haveS(b, x) ∼ CxwhereS(b, x) = # {b : |b| ≤ x and db is squarefree} .Thus the set of integers b for which db is squarefree has positive density.32 M. LAVALLEE, B. SPEARMAN AND Q. YANGWe close this section with some examples of intersective polynomials ob-tained from our theorem.Example 1. For the following values of b we construct intersective polyno-mials 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.b db Intersective Polynomial−9 817 (x5 − 2x4 − 7x3 + 17x2 − 9x+ 1)(x2 − 817)−8 401 (x5 − 2x4 − 6x3 + 15x2 − 8x+ 1)(x2 − 401)0 −47 (x5 − 2x4 + 2x3 − x2 + 1)(x2 + 47)2 −239 (x5 − 2x4 + 4x3 − 5x2 + 2x+ 1)(x2 + 239).4. Related ExamplesFor 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 quinticfields 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 ramifiedprimes and associated decomposition groups, algorithms for factoring idealsare given in Cohen [4]. The quadratic subfield of the splitting field of f(x)is 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 above5 in the splitting field of f(x) and the decomposition group of this ideal isall 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.INTERSECTIVE D5 POLYNOMIALS 33References[1] S¸. Alaca and K. S. Williams, Introductory Algebraic Number Theory, Cambridge Uni-versity 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. Erdo¨s, Arithmetic properties of polynomials, J. London Math. Soc. 28 (1953), 416-425.[6] C. Hooley, Applications of sieve methods to the theory of numbers. Cambridge Tracts inMathematics, No. 70. Cambridge University Press, Cambridge-New York-Melbourne,1976.[7] M. J. Lavallee, B. K. Spearman, K. S. Williams and Q. Yang, Dihedral Quintic FieldsWith A Power Basis, Math. J. Okayama Univ. 47, (2005), 75-79.[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. LavalleeDepartment of Mathematics and StatisticsUniversity of British Columbia OkanaganKelowna, BC, Canada, V1V 1V7.e-mail address: Melisa Lavallee@hotmail.comBlair K. SpearmanDepartment of Mathematics and StatisticsUniversity of British Columbia OkanaganKelowna, BC, Canada, V1V 1V7.e-mail address: blair.spearman@ubc.caQiduan YangDepartment of Mathematics and StatisticsUniversity of British Columbia OkanaganKelowna, BC, Canada, V1V 1V7.e-mail address: qiduan.yang@ubc.ca(Received November 15, 2011 )(Revised April 10, 2012 )

Abstract. We give an infinite family of intersective polynomials with Galois group D 5 , the dihedral group of order 10

Melisa J Lavallee

Blair K Spearman

Qiduan Yang

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INTERSECTIVE POLYNOMIALS WITH GALOIS GROUP D 5

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INTERSECTIVE POLYNOMIALS WITH GALOIS GROUP D<sub>5</sub>

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