Math. J. Okayama Univ. 47 (2005), 75–79

DIHEDRAL QUINTIC FIELDS WITH A POWER BASIS
Melissa J. LAVALLEE, Blair K. SPEARMAN, Kenneth S. WILLIAMS
and Qiduan YANG
Abstract. It is shown that there exist inﬁnitely many dihedral quintic
ﬁelds with a power basis.

1. Introduction
Let K be an algebraic number ﬁeld of degree n. Let OK denote thering of integers of K. The ﬁeld K is said to possess a power basis if there
exists an element θ ∈ OK such that OK = Z + Zθ+ · · · + Zθn−1 . A ﬁeldhaving a power basis is called monogenic. Every quadratic ﬁeld is monogenic.
Dedekind [3] gave an example of a cubic ﬁeld which is not monogenic. If
K is a cyclic cubic ﬁeld Gras [7], [8] and Archinard [1] have given necessary
and suﬃcient conditions for K to be monogenic. Dummit and Kisilevsky[4] have shown that there exist inﬁnitely many cyclic cubic ﬁelds which
are monogenic. The same has been shown for non-cyclic cubic ﬁelds, purequartic ﬁelds, bicyclic quartic ﬁelds, dihedral quartic ﬁelds by Spearman andWilliams [15], Funakura [6], Nakahara [14], Huard, Spearman and Williams[10] respectively. It is not known if there are inﬁnitely many monogenic
cyclic quartic ﬁelds. If K is a cyclic ﬁeld of prime degree p ≥ 5 then Gras [9]has proved that K is monogenic if and only if K is the maximal real subﬁeld
of a cyclotomic ﬁeld. In particular there is only one monogenic cyclic quintic
ﬁeld.
In this paper we exhibit inﬁnitely many monogenic dihedral quintic ﬁelds.
After giving some preliminary results in Section 2, we prove the following
theorem in Section 3.
Theorem. There are inﬁnitely many integers b such that the quintic ﬁelds
Q(θ), θ5 − 2θ4 + (b + 2)θ3 − (2b + 1)θ2 + bθ + 1 = 0,
are distinct, dihedral and monogenic.
2. A parametric family of quintics
For an integer b we deﬁne
Fb (x) := x5 − 2x4 + (b + 2)x3 − (2b + 1)x2 + bx + 1,b ∈ Z.

Mathematics Subject Classiﬁcation. 11R21.
Key words and phrases. Dihedral quintic ﬁeld, power basis, monogenic.
The second, third and fourth authors were supported by research grants from theNatural Sciences and Engineering Research Council of Canada.
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M. J. LAVALLEE, B. K. SPEARMAN, K. S. WILLIAMS AND Q. YANG

As x5 + x2 + 1 and x5 + x3 + x2 + x + 1 are irreducible (mod 2), we have
Lemma 2.1. Fb (x) is irreducible for all b ∈ Z.
Using MAPLE we ﬁnd
Lemma 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 ∈ 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 quadratic
subﬁeld of the splitting ﬁeld of Fb (x) is(√)
Q−4b3 − 28b2 − 24b − 47 .
3. Proof of theorem
By a theorem of Erdös [5] there are inﬁnitely many integers b such that
For each such b let θb be a root ofFb (x) = 0 and set Kb = Q (θb ). By Lemma 2.3 each Kb is a dihedral quintic
ﬁeld. The discriminant d(Kb ) of Kb is given by
4b3 + 28b2 + 24b + 47 is squarefree.

d(Kb ) = d2b fb4 ,where
db = discriminant of the quadratic subﬁeld of the splitting ﬁeld of Fb (x)andfb = conductor of Kb ∈ N,
see [13, p. 836]. By Lemma 2.4 we havedb = −4b3 − 28b2 − 24b − 47so thatd(Kb ) = (4b3 + 28b2 + 24b + 47)2 fb4 .

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By 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, θb , θb2 , θb3 , θb4 ) and so is monogenic.
As
4k 3 + 28k 2 + 24k + 47 = ±(4b3 + 28b2 + 24b + 47)
has at most six solutions for a given integer b, we can pick an inﬁnite subsequence of the original sequence of b’s for which 4b3 + 28b2 + 24b + 47 issquarefree in such a way that all the ﬁelds Kb are distinct.
¤2If 4b3 + 28b+ 47 }is squarefree the dihedral quintic ﬁeld Kb has the{ + 24bpower basis 1, θ, θ2 , θ3 , θ4{ , where we have} written θ for θb . In addition Kb2
34
also has the power bases 1, φ, φ , φ , φ with

φ1 = bθ − (b + 1)θ2 + θ3 − θ4and
φ2 = (2b + 1)θ − (b + 2)θ2 + 2θ3 − θ4 .
This follows as the minimal polynomials of φ1 and φ2 are by MAPLE
x5 + x4 + (b + 3)x3 + (b + 4)x2 + 3x + 1
andx5 − 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, φ, φ2 , φ3 , φ4given by
φ1 = θ3 − θ4 ,φ2 = 2θ − 2θ2 + 2θ3 − θ4 ,φ3 = θ + θ3 ,φ4 = θ − 2θ2 + θ3 ,φ5 = 6θ − 7θ2 + 5θ3 − 2θ4 ,φ6 = θ2 − θ3 ,φ7 = θ − θ2 + θ3 ,φ8 = θ − θ2 .

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M. J. LAVALLEE, B. K. SPEARMAN, K. S. WILLIAMS AND Q. YANG

We 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 estmonogéne, Enseignement Math. 20 (1974), 179-203.
[2] A. Brumer, preprint.
[3] R. Dedekind, Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Kongruenzen, Abh. Kgl. Ges. Wiss. Göttingen 23 (1878), 1-23.
[4] D. S. Dummit and H. Kisilevsky, Indices in cyclic cubic ﬁelds, in “Number Theoryand Algebra”, Academic Press, 1977, 29-42.
[5] P. Erdös, Arithmetic properties of polynomials, J. London Math. Soc. 28 (1953), 416425.
[6] T. Funakura, On integral bases of pure quartic ﬁelds, Math. J. Okayama Univ. 26(1984), 27-41.
[7] M.-N. Gras, Sur les corps cubiques cycliques dont l’anneau des entiers monogène, Ann.
Sci. Univ. Besançon 3, No. 6, 26 pp, 1973.
[8] M.-N. Gras, Lien entre le groupe des unités et la monogéneite des corps cubiques
cycliques, Ann. Sci. Univ. Besançon No. 1, 19 pp, 1975-76.
[9] M.-N. Gras, Non monogénéité de l’anneau des entiers des extensions cycliques de Q
de degré 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 ﬁelds
with quadratic subﬁelds, 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 unramiﬁed extensions over quadratic ﬁelds, 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), 831847.
[14] T. Nakahara, On the indices and integral bases of non-cyclic but abelian biquadraticﬁelds, Arch. Math. 41 (1983), 504-508.
[15] B. K. Spearman and K. S. Williams, Cubic ﬁelds with a power basis, Rocky Mountain
J. Math. 31 (2001), 1103-1109.

Melissa J. Lavallee
Department of Mathematics and Statistics,
Okanagan University College,
Kelowna, B.C. Canada V1V 1V7

Blair K. Spearman
Department of Mathematics and Statistics,
Okanagan University College,
Kelowna, B.C. Canada V1V 1V7e-mail address: BSpearman@ouc.bc.ca

DIHEDRAL QUINTIC FIELDS

Kenneth S. Williams
School of Mathematics and Statistics,
Carleton University,Ottawa, Ontario, Canada K1S 5B6e-mail address: williams@math.carleton.ca
Qiduan Yang
Department of Mathematics and Statistics,
Okanagan University College,Kelowna, B.C. Canada V1V 1V7(Received September 24, 2004 )(Revised January 12, 2005 )79