It is shown that there exist infinitely many dihedral quintic fields with a power basis.</p

Lavallee, Melissa J.

Spearman, Blair K.

Williams, Kenneth S.

Yang, Qiduan

English

It is shown that there exist infinitely many dihedral quintic fields with a power basis

Melissa J. Lavallee

Blair K. Spearman

Kenneth S. Williams

Qiduan Yang

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Dihedral quintic fields with a power basis

Okayama University Scientific Achievement Repository

Mathematical Journal of OkayamaUniversityVolume 47, Issue 1 2005 Article 6JANUARY 2005Dihedral Quintic Fields with a Power BasisMelissa J. Lavallee∗ Blair K. Spearman†Kenneth S. Williams‡ Qiduan Yang∗∗∗Okanagan University College†Okanagan University College‡Carleton University∗∗Okanagan University CollegeCopyright c©2005 by the authors. Mathematical Journal of Okayama University is produced byThe Berkeley Electronic Press (bepress). http://escholarship.lib.okayama-u.ac.jp/mjouDihedral Quintic Fields with a Power BasisMelissa J. Lavallee, Blair K. Spearman, Kenneth S. Williams, and Qiduan YangAbstractIt is shown that there exist infinitely many dihedral quintic fields with a power basis.KEYWORDS: Dihedral quintic field, power basis, monogenicMath. J. Okayama Univ. 47 (2005), 75{79DIHEDRAL QUINTIC FIELDS WITH A POWER BASISMelissa J. LAVALLEE, Blair K. SPEARMAN, Kenneth S. WILLIAMSand Qiduan YANGAbstract. It is shown that there exist in¯nitely many dihedral quintic¯elds with a power basis.1. IntroductionLet K be an algebraic number ¯eld of degree n. Let OK denote thering of integers of K. The ¯eld K is said to possess a power basis if thereexists an element µ 2 OK such that OK = Z + Zµ+ ¢ ¢ ¢+ Zµn¡1. A ¯eldhaving a power basis is called monogenic. Every quadratic ¯eld is monogenic.Dedekind [3] gave an example of a cubic ¯eld which is not monogenic. IfK is a cyclic cubic ¯eld Gras [7], [8] and Archinard [1] have given necessaryand su±cient conditions for K to be monogenic. Dummit and Kisilevsky[4] have shown that there exist in¯nitely many cyclic cubic ¯elds whichare monogenic. The same has been shown for non-cyclic cubic ¯elds, purequartic ¯elds, bicyclic quartic ¯elds, dihedral quartic ¯elds by Spearman andWilliams [15], Funakura [6], Nakahara [14], Huard, Spearman and Williams[10] respectively. It is not known if there are in¯nitely many monogeniccyclic quartic ¯elds. If K is a cyclic ¯eld of prime degree p ¸ 5 then Gras [9]has proved that K is monogenic if and only if K is the maximal real sub¯eldof a cyclotomic ¯eld. In particular there is only one monogenic cyclic quintic¯eld.In this paper we exhibit in¯nitely many monogenic dihedral quintic ¯elds.After giving some preliminary results in Section 2, we prove the followingtheorem in Section 3.Theorem. There are in¯nitely many integers b such that the quintic ¯eldsQ(µ); µ5 ¡ 2µ4 + (b+ 2)µ3 ¡ (2b+ 1)µ2 + bµ + 1 = 0;are distinct, dihedral and monogenic.2. A parametric family of quinticsFor an integer b we de¯neFb(x) := x5 ¡ 2x4 + (b+ 2)x3 ¡ (2b+ 1)x2 + bx+ 1; b 2 Z:Mathematics Subject Classi¯cation. 11R21.Key words and phrases. Dihedral quintic ¯eld, power basis, monogenic.The second, third and fourth authors were supported by research grants from theNatural Sciences and Engineering Research Council of Canada.751Lavallee et al.: Dihedral Quintic Fields with a Power BasisProduced by The Berkeley Electronic Press, 200576 M. J. LAVALLEE, B. K. SPEARMAN, K. S. WILLIAMS AND Q. YANGAs x5 + x2 + 1 and x5 + x3 + x2 + x+ 1 are irreducible (mod 2), we haveLemma 2.1. Fb(x) is irreducible for all b 2 Z.Using MAPLE we ¯ndLemma 2.2. disc(Fb(x)) = (4b3 + 28b2 + 24b+ 47)2.We note that the cubic polynomial 4b3+28b2+24b+47 is irreducible. Thepolynomial Fb(x) is a special case of the polynomial Ra;b(x) (a; b 2 Z) givenbyRa;b(x) = x5 + (a¡ 3)x4 + (b¡ a+ 3)x3 + (a2 ¡ a¡ 1¡ 2b)x2 + bx+ a;which was studied by Brumer [2] and Kondo [12]. Our polynomial Fb(x) isobtained by setting a = 1. It is shown in [11, pp. 44-46] that the Ra;b forma generic dihedral family and it is known when the Galois group of Ra;b iscyclic of order 5. From this work we have the following two lemmas.Lemma 2.3.Gal(Fb(x)) = Z5; if ¡ (4b3 + 28b2 + 24b+ 47) is a square in Z:Gal(Fb(x)) = D5; if ¡ (4b3 + 28b2 + 24b+ 47) is not a square in Z:Lemma 2.4. If ¡(4b3 + 28b2 + 24b+ 47) 6= square in Z then the quadraticsub¯eld of the splitting ¯eld of Fb(x) isQ³p¡4b3 ¡ 28b2 ¡ 24b¡ 47´:3. Proof of theoremBy a theorem of ErdÄos [5] there are in¯nitely many integers b such that4b3 + 28b2 + 24b + 47 is squarefree. For each such b let µb be a root ofFb(x) = 0 and set Kb = Q (µb). By Lemma 2.3 each Kb is a dihedral quintic¯eld. The discriminant d(Kb) of Kb is given byd(Kb) = d2bf4b ;wheredb = discriminant of the quadratic sub¯eld of the splitting ¯eld of Fb(x)andfb = conductor of Kb 2 N;see [13, p. 836]. By Lemma 2.4 we havedb = ¡4b3 ¡ 28b2 ¡ 24b¡ 47so thatd(Kb) = (4b3 + 28b2 + 24b+ 47)2f4b :2Mathematical Journal of Okayama University, Vol. 47 [2005], Iss. 1, Art. 6http://escholarship.lib.okayama-u.ac.jp/mjou/vol47/iss1/6DIHEDRAL QUINTIC FIELDS 77By Lemma 2.2 we havedisc(Fb(x)) = (4b3 + 28b2 + 24b+ 47)2:As d(Kb) divides disc(Fb(x)), we deduce that fb = 1 so thatd(Kb) = disc(Fb(x)) = §(4b3 + 28b2 + 24b+ 47)2:Hence Kb has a power basis (namely©1; µb; µ2b ; µ3b ; µ4bª) and so is monogenic.As4k3 + 28k2 + 24k + 47 = §(4b3 + 28b2 + 24b+ 47)has at most six solutions for a given integer b, we can pick an in¯nite sub-sequence of the original sequence of b's for which 4b3 + 28b2 + 24b + 47 issquarefree in such a way that all the ¯elds Kb are distinct. ¤If 4b3+28b2+24b+47 is squarefree the dihedral quintic ¯eld Kb has thepower basis©1; µ; µ2; µ3; µ4ª, where we have written µ for µb. In addition Kbalso has the power bases©1; Á; Á2; Á3; Á4ªwithÁ1 = bµ ¡ (b+ 1)µ2 + µ3 ¡ µ4andÁ2 = (2b+ 1)µ ¡ (b+ 2)µ2 + 2µ3 ¡ µ4:This follows as the minimal polynomials of Á1 and Á2 are by MAPLEx5 + x4 + (b+ 3)x3 + (b+ 4)x2 + 3x+ 1andx5 ¡ 4bx4 + (6b2 ¡ 2b¡ 1)x3 + (¡4b3 + 6b2 + 4b+ 2)x2+ (b4 ¡ 6b3 ¡ 5b2 ¡ 4b¡ 2)x+ (2b4 + 2b3 + 2b2 + 2b+ 1)respectively, each of discriminant (4b3 + 28b2 + 24b+ 47)2.When b = 0, we have the additional eight power bases©1; Á; Á2; Á3; Á4ªgiven byÁ1 = µ3 ¡ µ4;Á2 = 2µ ¡ 2µ2 + 2µ3 ¡ µ4;Á3 = µ + µ3;Á4 = µ ¡ 2µ2 + µ3;Á5 = 6µ ¡ 7µ2 + 5µ3 ¡ 2µ4;Á6 = µ2 ¡ µ3;Á7 = µ ¡ µ2 + µ3;Á8 = µ ¡ µ2:3Lavallee et al.: Dihedral Quintic Fields with a Power BasisProduced by The Berkeley Electronic Press, 200578 M. J. LAVALLEE, B. K. SPEARMAN, K. S. WILLIAMS AND Q. YANGWe do not know if there are any more power bases when b = 0.References[1] G. Archinard, Extensions cubiques cycliques de Q dont l'anneau des entiers estmonog¶ene, Enseignement Math. 20 (1974), 179-203.[2] A. Brumer, preprint.[3] R. Dedekind, ÄUber den Zusammenhang zwischen der Theorie der Ideale und der The-orie der hÄoheren Kongruenzen, Abh. Kgl. Ges. Wiss. GÄottingen 23 (1878), 1-23.[4] D. S. Dummit and H. Kisilevsky, Indices in cyclic cubic ¯elds, in \Number Theoryand Algebra", Academic Press, 1977, 29-42.[5] P. ErdÄos, Arithmetic properties of polynomials, J. London Math. Soc. 28 (1953), 416-425.[6] T. Funakura, On integral bases of pure quartic ¯elds, Math. J. Okayama Univ. 26(1984), 27-41.[7] M.-N. Gras, Sur les corps cubiques cycliques dont l'anneau des entiers monogµene, Ann.Sci. Univ. Besan»con 3, No. 6, 26 pp, 1973.[8] M.-N. Gras, Lien entre le groupe des unit¶es et la monog¶eneite des corps cubiquescycliques, Ann. Sci. Univ. Besan»con No. 1, 19 pp, 1975-76.[9] M.-N. Gras, Non monog¶en¶eit¶e de l'anneau des entiers des extensions cycliques de Qde degr¶e premier l ¸ 5, J. Number Theory 23 (1986), 347-353.[10] J. G. Huard, B. K. Spearman and K. S. Williams, Integral bases for quartic ¯eldswith quadratic sub¯elds, J. Number Theory 51 (1995), 87-102.[11] C. U. Jensen, A. Ledet and N. Yui, Generic Polynomials, Constructive Aspects ofthe Inverse Galois Problem, Mathematical Sciences Research Institute Publications,Cambridge University Press, 2002.[12] T. Kondo, Some examples of unrami¯ed extensions over quadratic ¯elds, Sci. Rep.Tokyo Woman's Christian Univ., No. 120-121 (1977), 1399-1410.[13] D. C. Meyer, Multiplicities of dihedral discriminants, Math. Comp. 58 (1992), 831-847.[14] T. Nakahara, On the indices and integral bases of non-cyclic but abelian biquadratic¯elds, Arch. Math. 41 (1983), 504-508.[15] B. K. Spearman and K. S. Williams, Cubic ¯elds with a power basis, Rocky MountainJ. Math. 31 (2001), 1103-1109.Melissa J. LavalleeDepartment of Mathematics and Statistics,Okanagan University College,Kelowna, B.C. Canada V1V 1V7Blair K. SpearmanDepartment of Mathematics and Statistics,Okanagan University College,Kelowna, B.C. Canada V1V 1V7e-mail address: BSpearman@ouc.bc.ca4Mathematical Journal of Okayama University, Vol. 47 [2005], Iss. 1, Art. 6http://escholarship.lib.okayama-u.ac.jp/mjou/vol47/iss1/6DIHEDRAL QUINTIC FIELDS 79Kenneth S. WilliamsSchool of Mathematics and Statistics,Carleton University,Ottawa, Ontario, Canada K1S 5B6e-mail address: williams@math.carleton.caQiduan YangDepartment of Mathematics and Statistics,Okanagan University College,Kelowna, B.C. Canada V1V 1V7(Received September 24, 2004 )(Revised January 12, 2005 )5Lavallee et al.: Dihedral Quintic Fields with a Power BasisProduced by The Berkeley Electronic Press, 2005

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Dihedral Quintic Fields with a Power Basis

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