Math. J. Okayama Univ. 59 (2017), 113–116

A REMARK ON THE
LAVALLEE-SPEARMAN-WILLIAMS-YANG FAMILYOF QUADRATIC FIELDS
Kwang-Seob Kim and Yasuhiro Kishi
Abstract. In [4], M. J. Lavallee, B. K. Spearman, K. S. Williams andQ. Yang introduced a certain parametric D5 -quintic polynomial andstudied its splitting field. The present paper gives an infinite family of
quadratic fields with class number divisible by 5 by using properties ofits polynomial.

1. Introduction
One of the main topics in algebraic number theory is to study the structureof ideal class groups of number fields, particularly the divisibility of classnumbers. In the case of quadratic fields, numerous related results are knownby many authors. As was first shown by T. Nagell [7] (resp. by Y. Yamamoto[11]), there exist infinitely many imaginary (resp. real) quadratic fields withclass numbers divisible by an arbitrary given integer n. After that, several
authors gave such results (for example, [1], [10], [6], [9], [12], [5], and so on).
Every unramified cyclic extension of prime degree p of a quadratic field isnormal over Q and is a Dp -extension of Q, that is, a Galois extension whoseGalois group is isomorphic to the dihedral group Dp of order 2p. Therefore,
it is given by roots of a polynomial of degree p with rational coefficients.
Y. Kishi and K. Miyake [2] used some D3 -cubic polynomials to classify all
quadratic fields whose class number is divisible by 3. Here, a D3 -cubic
polynomial means a polynomial of degree 3 such that its minimal splitting
field over Q is a D3 -extension of Q. In the case p = 5, Kishi [3] found afamily of D5 -quintic polynomials such that its minimal splitting fields over
Q are unramified cyclic quintic extensions of quadratic subfields. But hismethods is slightly complicated. In this paper, we will give such polynomials
that are simpler and more efficient by using M. J. Lavallee, B. K. Spearman,K. S. Williams and Q. Yang’s results.
For a number field K, let D(K) denote the discriminant of K. For apolynomial f , let d(f ) (resp. SplQ (f )) denote the discriminant (resp. theminimal splitting field over Q) of f .
Mathematics Subject Classification. Primary 11R29; Secondary 11R11.
Key words and phrases. Class numbers, Quadratic fields, D5 -polynomials.
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K.-S. KIM AND Y. KISHI

2. Main Theorem
We need the following well-known lemma before we state our main theorem.
Lemma 1 (Dedekind). Let F be an algebraic number field and D be thedifferent of F over Q. Let p be a prime divisor in F over a prime numberp, and pd k D and pe k p. Then(a) if p - e, then d = e − 1,(b) if pv k e (v > 0), then e ≤ d ≤ ev + e − 1.
Proof. See, for example, [8, Chap. III, §2, (2.6) Theorem].

Theorem. Let f be a D5 -quintic polynomial and put L := SplQ (f ). Moreover, let K be the unique quadratic subfield of L. Suppose that56 - d(f ) and q 4 - d(f )for every prime number q 6= 5. Then L is an unramified cyclic quintic
extension of K. Especially, the class number of K is divisible by 5.
Proof. Let F = Q(θ) where f (θ) = 0. Then we haved(f ) = a2 D(F )for some integer a.
If L/K is unramified, then the class number of K is divisible by 5 by classfield theory. So it suffices to show that if L/K is not unramified, then wehave
56 | D(F ) or q 4 | D(F )for some prime number q 6= 5. Suppose that p is a prime ideal of K whichis ramified in L/K.
Case 1, where 5 | p. Since p is ramified in L/K and [L : K] = 5, 5 iswildly ramified in F/Q. By Lemma 1 (b), it must hold that 5 ≤ d ≤ 9.
This implies that 55 | D(F ). Now we may view Gal(L/K) as a subgroup of
the symmetric group S5 of order 5 because f is a quintic polynomial. Since
Gal(L/K) ' D5 is contained in the alternative group A5 of degree 5, D(F )must be a square in Z. Thus we have 56 | D(F ).
Case 2, where q | p with q ≡ 1 mod 5. Since p is ramified in L/K and[L : K] = 5, q is tamely ramified in F/Q. By Lemma 1 (a), we have q 4 k D.
In this case, we can easily verify that q 4 | D(F ). The proof is completed. 
3. The Lavallee-Spearman-Williams-Yang family of quadratic
fieldsBy using our theorem, we will give an infinite family of quadratic fields
with class number divisible by 5. This is based on Lavallee, Spearman,
Williams and Yang’s works [4]. Let us refer to them.

THE LAVALLEE-SPEARMAN-WILLIAMS-YANG FAMILY115

For an integer b, we definefb (x) := x5 − 2x4 + (b + 2)x3 − (2b + 1)x2 + bx + 1.

First they proved that fb is irreducible for any b ∈ Z andd(fb ) = (4b3 + 28b2 + 24b + 47)2(see [4, Lemmas 2.1 and 2.2]). They also proved the following.
Proposition 1 ([4, Lemmas 2.3 and 2.4]). If −(4b3 + 28b2 + 24b + 47) is
not a square in Z, then SplQ (fb ) is a D5 -extension of Q and the uniquequadratic subfield of SplQ (fb ) isp
Q( −(4b3 + 28b2 + 24b + 47)).
Furthermore, there are infinitely many integers b such that −(4b3 +28b2 +24bp+ 47) is square-free, is not a square in Z, and the quadratic fields
Q( −(4b3 + 28b2 + 24b + 47)) are distinct, as we can see in the proof of[4, Theorem]. From this, together with Proposition 1 and our main theorem, the following proposition is immediately obtained.
Proposition 2. There exist infinitely many quadratic fields of the formpQ( −(4b3 + 28b2 + 24b + 47)), b ∈ Z

whose class number is divisible by 5.

Acknowledgement
The authors would like to thank the referee for his careful reading of themanuscript and for giving many valuable comments.
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Kwang-Seob Kim
School of Mathematics
Korea Institute for Advanced StudySeoul, 130-722 Korea
e-mail address: kwang12@kias.re.kr
Yasuhiro Kishi
Department of Mathematics
Aichi University of Education
Aichi, 448-8542 Japane-mail address: ykishi@auecc.aichi-edu.ac.jp(Received April 21, 2016 )(Accepted August 22, 2016 )