Fundamental Units and Regulators of an Infinite Family of Cyclic Quartic Function Fields

Home , Quartic function, Quintic function

J. Korean Math. Soc. 54 (2017), No. 2, pp. 417–426 https://doi.org/10.4134/JKMS.j160002 pISSN: 0304-9914 / eISSN: 2234-3008

FUNDAMENTAL UNITS AND REGULATORS OF AN INFINITE FAMILY OF CYCLIC QUARTIC FUNCTION FIELDS

Jungyun Lee and Yoonjin Lee

Abstract. We explicitly determine fundamental units and regulators of an infinite family of cyclic quartic function fields Lh of unit rank 3 with a parameter h in a polynomial ring Fq[t], where Fq is the finite field of order q with characteristic not equal to 2. This result resolves the second part of Lehmer’s project for the function field case.

1. Introduction Lecacheux [9, 10] and Darmon [3] obtain a family of cyclic quintic fields over Q, and Washington [22] obtains a family of cyclic quartic fields over Q by using coverings of modular curves. Lehmer’s project [13, 14] consists of two parts; one is finding families of cyclic extension fields, and the other is computing a system of fundamental units of the families. Washington [17, 22] computes a system of fundamental units and the regulators of cyclic quartic fields and cyclic quintic fields, which is the second part of Lehmer’s project. We are interested in working on the second part of Lehmer’s project for the families of function fields which are analogous to the type of the number field families produced by using modular curves given in [22]: that is, finding a system of fundamental units and regulators of families of cyclic extension fields over the rational function field Fq(t). In [11], we obtain the results for the quintic extension case. In this paper, we work on the quartic extension case; that is, we explicitly determine a system of fundamental units and regulators of the following infinite family of quartic function fields Lh over Fq(t). { }

Received January 4, 2016. 2010 Mathematics Subject Classification. 11R29, 11R58. Key words and phrases. regulator, function field, quintic extension. The authors were supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(2009-0093827), the first named author was also supported by the National Research Foundation of Ko- rea(NRF) grant founded by the Korea government(2011-0023688), and the second named author by the National Research Foundation of Korea(NRF) grant founded by the Korea government(MEST)(2014-002731).

c 2017 Korean Mathematical Society 417 418 J. LEE AND Y. LEE

Let k = Fq(t) be a rational function ﬁeld and Lh = k(αh) be a quartic extension over k generated by a root αh of 4 2 3 3 2 2 2 Fh(x)= x h x (h +2h +4h + 2)x h x +1, − − − 2 where h is a monic polynomial in Fq[t] such that h(h + 2)(h + 4) is square free in Fq[t]. Then we show that Lh = k(αh) is a real cyclic function ﬁeld of unit rank three, and we explicitly determine a system of fundamental units and regulators of Lh as the following main theorem.

2 Theorem 1.1. Let h be a monic polynomial in Fq[t] such that h(h + 2)(h + 4) is square free in Fq[t]. The regulator R(Lh) of Lh is explicitly given by 3 R(Lh) = 10(deg h) .

Furthermore, a system of the fundamental units of Lh are αh, σ(αh),ǫh with { } the following unit group of Lh ∗ U(Lh)= F αh, σ(αh),ǫh , q × h i 2 where ǫh = h + √h +4 and σ is a generator of the Galois group Gal(Lh/k).

2. Preliminary

Let Lh and Fh(x) be the same as given in Section 1. Then all four roots αh,1, αh,2, αh,3, αh,4 of Fh(x) are as follows:

h2 + (h + 2)√h2 +4 2h(h + 2)(h2 +4)+2h2(h + 2)√h2 +4 ± , q 4 h2 (h + 2)√h2 +4 2h(h + 2)(h2 + 4) 2h2(h + 2)√h2 +4 − ± − . q 4 2 Let Kh = k(√h + 4). Then Kh is a unique quadratic subﬁeld of Lh and the 2 fundamental unit ǫh of Kh is h + √h +4. It is known that Lh is a cyclic extension over k with 2 Gal(Lh/k)= σ , and Gal(Lh/Kh)= σ , h i h i where σ is deﬁned by

1 3 2 3 σ(αh) = (h + ) (h + h +3h + )αh h +2 − h +2 3 1 + ( h2 + h 2+ )α2 + (1 )α3 . − − h +2 h − h +2 h

We have Lagrange resolvent r1 = αh,1 + αh,2i αh,3 αh,4i for Lh, where i2 = 1, and we ﬁnd − − − 4 2 2 2 2 1 = r = h (h + 2) (h + 4)(h 2i) Fq(i)(t). R 1 − ∈ REGULATORS OF AN INFINITE FAMILY OF QUARTIC FUNCTION FIELDS 419

2 We notice that the discriminant DKh of Kh is h + 4 and primes in k dividing h and h + 2 are ramiﬁed in Lh; hence, according to conductor-discriminant formula, it follows that the discriminant DLh of Lh over k is given by 2 2 2 3 DLh = h (h + 2) (h + 4) .

Proposition 2.1. The inﬁnite prime ℘∞ of k splits completely in Lh; so Lh has unit rank 3, and Lh is a real function ﬁeld.

4 Proof. Let P∞ be an inﬁnite prime of k(√ 1) lying over ℘∞ k and℘ ˜∞ R 4 ∈ (resp. P˜ ) be the inﬁnite prime of k(i) (resp. k(i, √ 1)) lying over ℘ (resp. ∞ R ∞ P∞). Then we have F −1 4 F 4 −1 k(i)℘˜∞ = q(i)((t )) and k(i, 1) ˜ = q(i, 1)((t )). R P∞ R 2 2 2 2 −1 If we express 1 = h (h + 2) (h + 4)(ph 2i) in Fq(i)((pt )), we have R − 8 8d 1 = a t + lower terms on t, R d d i F F∗ where h = i=0 ait for ai q, (i = 0, 1,...,d 1) and ad q . Thus we have ∈ − ∈ P 4 −1 1 Fq(i)((t )) R ∈ and p 4 F 4 −1 F −1 k(i, 1) ˜ = q(i, 1)((t )) = q(i)((t )) = k(i)℘˜∞ ; R P∞ R this implies thatp p 4 k( 1)P∞ = k℘∞ , R which completes the proof. p

The inﬁnite prime ℘∞ of k splits completely in Lh; so we have k Lh −1 ⊆ ⊆ k∞ = Fq((t )), where k∞ is the completion of k at ℘∞. For a nonzero element ∞ −i a = cit k with m Z, ci Fq(i m) and c m =0, we deﬁne i=−m ∈ ∞ ∈ ∈ ≥− − 6 P deg a = m. Let U(Lh) (resp. U(Kh)) be the unit group of the maximal order of Lh (resp. Kh). Let 2 ∗ U(Lh/Kh) := ǫ U(Lh) N (ǫ)= ǫ σ (ǫ) F . { ∈ | Lh/Kh · ∈ q } It is known [4] that there is ηh Lh with ∈ ∗ U(Lh/Kh)= F ηh, σ(ηh) , q × h i and we call ηh a relative fundamental unit of Lh over Kh. Let R(Lh) (resp. R(Kh)) be the regulator of Lh (resp. the regulator of Kh) and for ǫi (i =1, 2, 3) U(Lh), ∈ deg ǫ1 deg ǫ2 deg σ(ǫ3) 2 (ǫ1,ǫ2,ǫ3) := det deg σ(ǫ1) deg σ(ǫ2) deg σ (ǫ3) . R  2 2 3  deg σ (ǫ1) deg σ (ǫ2) deg σ (ǫ3)   Let DLh/Kh (resp. DLh/k) denote the discriminant of Lh over Kh (resp. Lh over k). 420 J. LEE AND Y. LEE

3. Determination of relative fundamental units

In this section, we show that the relative fundamental unit ηh of Lh over Kh is equal to a root αh of 4 2 3 3 2 2 2 Fh(x)= x h x (h +2h +4h + 2)x h x +1 − − − F∗ up to constant in q . It is known [4] that

QL := [U(Lh): U(Kh)U(Lh/Kh)] 1, 2 h ∈{ } and (ǫK , ηh, σ(ηh)) = QL R(Lh). R h h We note that for α U(Lh/Kh) and β U(Kh), we have ∈ ∈ (β, α, σ(α)) = 2 deg(β) (deg α)2 + (deg σ(α))2 . R Thus, for determination of R(Lh) and a relative unit ηh, we need a lower bound 2 2 and an upper bound of (deg ηh) + deg(σ(ηh)) .

Proposition 3.1. Let ηh Lh be such that ∈ ∗ U(Lh/Kh)= F ηh, σ(ηh) . q × h i Then we have 2 2 2 2 4.5(deg h) (deg ηh) + deg(σ(ηh)) 5(deg h) . ≤ ≤ Proof. Since αh U(Lh/Kh), we have for integers a,b ∈ a b αh = ηhσ(ηh) and 2 2 2 2 2 2 (deg αh) + (deg σ(αh)) = (a + b ) (deg ηh) + deg(σ(ηh)) . We note that 2 2 αh = h + h +1+ + h ··· and 1 2 σ(αh)= h 1 + + . − − − h h3 ··· Thus, we have deg αh = 2deg h and deg σ(αh) = deg h. Finally, we obtain that 2 2 2 2 2 (deg ηh) + deg(σ(ηh)) (deg αh) + (deg σ(αh)) = 5(deg h) . ≤ Now, we note that 2 2 2 2 3 DLh = NKh/k(DLh/Kh )DKh = h (h + 2) (h + 4) . 2 Since DKh = h + 4, we have that 2 2 2 NKh/k(DLh/Kh )= h (h + 2) (h + 4). REGULATORS OF AN INFINITE FAMILY OF QUARTIC FUNCTION FIELDS 421

Moreover, we have 2 2 D (ηh σ (ηh)) Lh/Kh | − and 2 2 N (D ) N (ηh σ (ηh)) . Kh/k Lh/Kh | Kh/k − Thus we have

2 2 2 2 2 3 2 h (h + 2) (h + 4) (ηh σ (ηh)) (σ(ηh) σ (ηh)) . | − − ∗ 2 3 We observe that for c1,c2 F , σ (ηh)= c1/ηh, σ (ηh)= c2/σ(ηh) ∈ q

deg (ηh c1/ηh)= deg ηh , and deg (σ(ηh) c2/σ(ηh)) = deg σ(ηh) ; − | | − | | thus, we have

2 2 2 deg h (h + 2) (h + 4) 2 deg ηh +2 deg σ(ηh) ≤ | | | | 1 2 2 2 2√2 (deg ηh) + (deg σ(ηh)) , ≤ so that we get

2 2 2 2 2 2 deg h (h + 2) (h + 4) 8 (deg ηh) + (deg σ(ηh)) . ≤ · Consequently, we obtain that

2 2 2 4.5(deg h) (deg ηh) + (deg σ(ηh)) . ≤

Theorem 3.2. A root αh of Fh(x) is a relative fundamental unit of Lh over F∗ Kh up to constant in q .

Proof. Since αh U(Lh/Kh), we have for integers a,b ∈ a b αh = ηhσ(ηh) and

2 2 2 2 2 2 (1) (deg αh) + (deg σ(αh)) = (a + b ) (deg ηh) + deg(σ(ηh)) . From (1) and Proposition 3.1, it follows that

5(deg h)2 4.5(a2 + b2)(deg h)2. ≥ Thus, we have a2 + b2 = 1; ±1 ±1 this implies that αh is ηh or σ(ηh) , which completes the proof. 422 J. LEE AND Y. LEE

4. Proof of the main result

In this section, we ﬁrst compute QLh , and we then complete the proof of Theorem 1.1. We need the following two lemmas. Lemma 4.1 is a criterion for determining whether QLh is 1 or not. A similar criterion in the number ﬁeld case is given in [4]. F∗ F∗ Lemma 4.1. Let U(Kh) = q ǫh and U(Lh/Kh) = q ηh, σ(ηh) . If 1−σ × h i ∗ × h i cǫhη is not a square in U(Lh) for any c F , then h ∈ q

QLh =1.

Proof. Suppose that QLh =1. Then we can take u U(Lh) U(Kh)U(Lh/Kh). 1+σ2 6 ∈ − As u U(Kh), we have ∈ 1+σ2 2λ+1 1+σ2 2λ ∗ u = c1ǫ or u = c2ǫ (c1,c2 F ). h h ∈ q σ2 1+σ2 2λ F∗ u u F∗ If u = c2ǫh (c2 q ), then ǫλ ǫλ q which implies that u ∈ h h ∈ ∈ 1+σ2 2λ+1 ∗ U(Lh/Kh)U(Lh/Kh). Thus we have u = c1ǫ (c1 F ); so we get h ∈ q u u σ2 c1ǫh = λ λ . ǫh ǫh u If we let u1 = ǫλ U(L), then we have h ∈ 1+σ2 ∗ c1ǫh = u (c1 F ). 1 ∈ q 1+σ Since u U(Lh/Kh), we have 1 ∈ 1+σ A B ∗ (2) u = c3η σ(ηh) (c3 F and A, B Z). 1 h ∈ q ∈ We note that if A and B have the same parity (that is, both are even or odd), then A+B −A+B 1+σ A B 2 ∗ η σ(ηh) = c4 η σ(ηh) 2 (c4 F ); h h ∈ q therefore, we get A+B −A+B 1+σ 2 ∗ u1/(η σ(ηh) 2 ) F ; h ∈ q so we have u1 U(Lh/Kh)U(Lh/Kh), which is a contradiction. This shows that A and B ∈have not the same parity. In other words, A 1 and B have same parity. Thus (2) implies that −

− A+B 1 −A+B+1 1+σ 2 ∗ u1/(η σ(ηh) 2 ) = c3ηh (c3 F ). h ∈ q A B− + 1 −A+B+1 2 2 1+σ ∗ Since (η σ(ηh) 2 ) F , by letting h ∈ q u1 u2 := A B− , + 1 −A+B+1 2 ηh σ(ηh) 2 REGULATORS OF AN INFINITE FAMILY OF QUARTIC FUNCTION FIELDS 423 we have 1+σ 1+σ2 ∗ (3) u = c3ηh and u = c4ǫh (c3,c4 F ); 2 2 ∈ q therefore, 1−σ 2 ∗ c5ǫhη = u for u2 U(Lh) and c5 F , h 2 ∈ ∈ q which completes the proof. 1−σ F∗ We ﬁrst show that cǫhηh is not square in U(Lm) for any c q . Then by Lemma 4.1 we get ∈

QLh =1. 3 It then follows that R(Lh) = 10(deg h) and ∗ U(Lh)= F αh, σ(αh),ǫh , q × h i 2 where ǫh = h + √h +4. 1−σ It is thus enough to show that cǫhηh is not square in U(Lh) for any F∗ 1−σ c q . To determine whether cǫhηh is a square in U(Lh) or not, we need the∈ following lemma. Lemma 4.2. Let E be a quadratic extension of F . If τ E is square in ∈ E, then NE/F (τ), T rE/F (τ)+2 NE/F (τ) and T rE/F (τ) 2 NE/F (τ) are square in F . − p p Proof. Using the following formulas in Proposition 3.1 in [15],

τ + NE/F (τ) √τ = and T rE/F (τp)+2 NE/F (τ) q τ NpE/F (τ) √τ = − , T r (τp) 2 N (τ) E/F − E/F the result follows immediately.q p Proof of Theorem 1.1. In Theorem 3.2, we ﬁnd that ∗ ηh = cαh (c F ). ∈ q ∗ Let τh = cǫhαh/σ(αh) (c F ). We note that ∈ q 2 2 NLh/Kh (cτh)= c ǫh, 3 2 h 1 2 T r (τh)+2 N (τh)= cǫh 2h h h(h + 2) h +4 , Lh/Kh Lh/Kh − − − 2 − 2 and q p 3 2 h 1 2 T r (τh) 2 N (τh)= cǫh 4 2h h h(h+2) h +4 . Lh/Kh − Lh/Kh − − − − 2 − 2 Let q p 3 2 h 1 2 δh,1 := cǫh 2h h h(h + 2) h +4 , − − − 2 − 2 p 424 J. LEE AND Y. LEE and 3 2 h 1 2 δh,2 := cǫh 4 2h h h(h + 2) h +4 . − − − − 2 − 2 Moreover, if δh,1 K h is square in Kh, then N (δh,p2) and T r (δh,1)+ ∈ Kh/k Kh/k 2 NKh/k(δh,1) and T rKh/k(δh,1) 2 NKh/k(δh,1) are square in k. We note that − p 2 4 p 3 2 N (δh,1)= c (4h + 16h + 32h + 64h + 64), Kh/k − which is not square in k for any c F∗. ∈ q Moreover, if δh,2 Kh is square in Kh, then N (δh,2) and T r (δh,2)+ ∈ Kh/k Kh/k 2 NKh/k(δh,2) and T rKh/k(δh,2) 2 NKh/k(δh,2) are square in k. We note that − p p 2 4 (4) N (δh,2)= 4c h . Kh/k − F∗ If 1 is not square in Ok/q, then (4) is not square in k for any c q . On the − 2 ∈ other hand, if 1 is square in Ok/q witha = 1 in Ok/q, then we have − − 2 4 2 3 2 2 2 T rKh/k(δh,2)+2 NKh/k(δh,2)= c(2a h +4a h + (8a +4a)h +8a h) and q 2 4 2 3 2 2 2 T r (δh,2) 2 N (δh,2)= c(2a h +4a h + (8a 4a)h +8a h); Kh/k − Km/k − both are not squareq in k for any c F∗. ∈ q Hence, from Lemma 4.2, it follows that δh,1 and δh,2 are not square in Kh and τh is not square in Lh. This completes the proof.

5. Infinitely many family of quartic function fields In this section, we show that there are infinitely many primes q such that 2 (h(t) + 4)(h(t) +2)h(t) is square free in Fq[t], where h(t) is a given monic polynomial in Z[t]. Consequently, Theorem 1.1 holds for infinitely many family of quartic function fields. Lemma 5.1. For a field K, a nonzero polynomial f(x) K[x] is square free if and only if f(x) is relatively prime to f ′(x) in K[x]. ∈ Proof. Let f(x) be a nonzero polynomial in K[x]. If f(x) is square free, then f(x) and f ′(x) have no common factors in K[x]; thus they are relatively prime. For the converse, if we assume that f(x) is not square free, then f(x) and f ′(x) have some common factor in K[x]; so f(x) and f ′(x) are not relatively prime in K[x]. Lemma 5.2. (1) Let K be a field and f(x) K[x] be a square free polynomial. Then for g(x) K[x], if f(g(x))∈is relatively prime to g′(x), then f(g(x)) is square free∈ in K[x]. (2) For f(x), g(x) Z[x], if f(g(x)) is square free in Q[x], then f(g(x)) ∈ ∈ Fq[x] is square free for every prime q, where f¯ (resp. g¯) denotes the reduction of coefficients of f (resp. g) modulo q. REGULATORS OF AN INFINITE FAMILY OF QUARTIC FUNCTION FIELDS 425

Proof. (1) It is suﬃcient to show that f(g(x)) and f ′(g(x))g′(x) are relatively prime by Lemma 5.1. Since f(x) is square free in K[x], f(x) and f ′(x) are relatively prime in K[x]; so f(g(x)) and f ′(g(x)) are also relatively prime in K[x]. Thus, if f(g(x)) and g′(x) are relatively prime by our assumption, then f(g(x)) is square free in K[x]. (2) If f(g(x)) is square free in Q[x], f(g(x)) and g′(x) are relatively prime in Q[x] by Lemma 5.1; hence, there exist h1(x) and h2(x) in Q[x] such that ′ f(g(x))h1(x)+ g (x)h2(x)=1. Thus we have ′ f¯(¯g(x))h¯1(x)+¯g (x)h¯2(x) = 1; equivalently, f¯(¯g(x)) andg ¯′(x) are relatively prime. It thus follows that f(g(x)) is square free in Fq[x] for every prime q from the part (1). k F 2 F∗ Theorem 5.3. Let h(t) be of the type t + c q[t] with (c + 4)(c + 2)c q . 2 ∈ ∈ Then (h(t) + 4)(h(t)+2)h(t) is square free in Fq[t] for any power q of a prime except q =2 or q dividing k. Proof. From Lemma 5.2(1), we obtain the result.

References [1] W. E. H. Berwick, Algebraic number-fields with two independent units, Proc. London Math. Soc. 34 (1932), 360–378. [2] J. W. S. Cassels, An Introduction to the Geometry of Numbers, Springer-Verlag, Berlin, 1971. [3] H. Darmon, Note on a polynomial of Emma Lehmer, Math. Comp. 56 (1991), no. 194, 795–800. [4] M.-N. Gras, Table numérique du nombre de classes et des unités des extensions cycliques réelles de degré4 de Q, Publ. Math. Besan¸con, 1977/1978, fasc. 2, pp. 1–26 & 1–53. [5] , Special units in real cyclic sextic fields, Math. Comp. 48 (1987), no. 177, 179– 182. [6] H. Hasse, Arithmetische Bestimmung von Grundeinheit und Klassenzahl in zyklischen kubischen und biquadratischen Zahlkorpern, Mathematische Abhandlungen, Band 3, Walter de Gruyter, Berlin (1975), 285–379, Originally published, 1950. [7] Y. Kishi, A family of cyclic cubic polynomials whose roots are systems of fundamental units, J. Number Theory 102 (2003), no. 1, 90–106. [8] A. J. Lazarus, On the class number and unit index of simplest quartic fields, Nagoya Math. J. 121 (1991), 1–13. [9] O. Lecacheux, Unités d′une famille de corps cycliques réeles de degré6 liés à la courbe modulaire X1(13), J. Number Theory 31 (1989), no. 1, 54–63. ′ [10] , Unités d une famille de corps liés à la courbe X1(25), Ann. Inst. Fourier (Greno- ble) 40 (1990), no. 2, 237–253. [11] J. Lee and Y. Lee, Fundamental units and regulators of quintic function fields, preprint. [12] Y. Lee, R. Scheidler, and C. Yarrish, Computation of the fundamental units and the regulator of a cyclic cubic function field, Experiment. Math. 12 (2003), no. 2, 211–225. [13] E. Lehmer, Connection between Gaussian periods and cyclic units, Math. Comp. 50 (1988), no. 182, 535–541. [14] D. H. Lehmer and E. Lehmer, The Lehmer project, Math. Comp. 61 (1993), no. 203, 313–317. 426 J. LEE AND Y. LEE

[15] S. R. Louboutin, Hasse unit indices of dihedral octic CM-fields, Math. Nachr. 215 (2000), 107–113. [16] , The simplest quartic fields with ideal class groups of exponents less than or equal to 2, J. Math. Soc. Japan 56 (2004), no. 3, 717–727. [17] R. Schoof and L. C. Washington, Quintic polynomials and real cyclotomic fields with large class numbers, Math. Comp. 50 (1988), no. 182, 543–556. [18] D. Shanks, The simplest cubic fields, Math. Comp. 28 (1974), 1137–1152. [19] Y. Y. Shen, Unit groups and class numbers of real cyclic octic fields, Trans. Amer. Math. Soc. 326 (1991), no. 1, 179–209. [20] A. Stein and H. C. Williams, Some methods for evaluating the regulator of a real quadratic function field, Experiment. Math. 8 (1999), no. 2, 119–133. [21] F. Thaine, Jacobi sums and new families of irreducible polynomials of Gaussian periods, Math. Comp. 70 (2001), no. 236, 1617–1640. [22] L. C. Washington, A family of cyclic quartic fields arising from modular curve, Math. Comp. 57 (1991), no. 196, 763–775. [23] H. C. Williams and C. R. Zarnke, Computer calculation of units in cubic fields, Proceed- ings of the Second Manitoba Confference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1972), pp. 433–468. Congressus Numerantium, No. VII, Utilitas Math., Winnipeg, Man., 1973. [24] Z. Zhang and Q. Yue, Fundamental units of real quadratic fields of odd class number, J. Number Theory 137 (2014), 122–129.

Jungyun Lee Institute of Mathematical Sciences Ewha Womans University Seoul 120-750, Korea E-mail address: [email protected]

Yoonjin Lee Department of Mathematics Ewha Womans University Seoul 120-750, Korea E-mail address: [email protected]