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

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1 J. Korean Mat. Soc. 54 (017), No., pp ttps://doi.org/ /jkms.j16000 pissn: / eissn: FUNDAMENTAL UNITS AND REGULATORS OF AN INFINITE FAMILY OF CYCLIC QUARTIC FUNCTION FIELDS Jungyun Lee Yoonjin Lee Abstract. We explicitly determine fundamental units regulators of an infinite family of cyclic quartic function fields L of unit rank 3 wit a parameter in a polynomial ring F q[t], were F q is te finite field of order q wit caracteristic not equal to. Tis result resolves te second part of Lemer s project for te function field case. 1. Introduction Lecaceux [9, 10] Darmon [3] obtain a family of cyclic quintic fields over Q, Wasington [] obtains a family of cyclic quartic fields over Q by using coverings of modular curves. Lemer s project [13, 14] consists of two parts; one is finding families of cyclic extension fields, te oter is computing a system of fundamental units of te families. Wasington [17, ] computes a system of fundamental units te regulators of cyclic quartic fields cyclic quintic fields, wic is te second part of Lemer s project. We are interested in working on te second part of Lemer s project for te families of function fields wic are analogous to te type of te number field families produced by using modular curves given in []: tat is, finding a system of fundamental units regulators of families of cyclic extension fields over te rational function field F q (t). In [11], we obtain te results for te quintic extension case. In tis paper, we work on te quartic extension case; tat is, we explicitly determine a system of fundamental units regulators of te following infinite family of quartic function fields {L } over F q (t). Received January 4, Matematics Subject Classification. 11R9, 11R58. Key words prases. regulator, function field, quintic extension. Te autors were supported by Basic Science Researc Program troug te National Researc Foundation of Korea(NRF) funded by te Ministry of Education( ), te first named autor was also supported by te National Researc Foundation of Korea(NRF) grant founded by te Korea government( ), te second named autor by te National Researc Foundation of Korea(NRF) grant founded by te Korea government(mest)( ). 417 c 017 Korean Matematical Society

2 418 J. LEE AND Y. LEE Let k = F q (t) be a rational function field L = k(α ) be a quartic extension over k generated by a root α of F (x) = x 4 x 3 ( )x x+1, were is a monic polynomial in F q [t] suc tat ( + )( + 4) is square free in F q [t]. Ten we sow tat L = k(α ) is a real cyclic function field of unit rank tree, we explicitly determine a system of fundamental units regulators of L as te following main teorem. Teorem 1.1. Let be a monic polynomial in F q [t] suc tat (+)( +4) is square free in F q [t]. Te regulator R(L ) of L is explicitly given by R(L ) = 10(deg) 3. Furtermore, a system of te fundamental units of L are {α,σ(α ),ǫ } wit te following unit group of L U(L ) = F q α,σ(α ),ǫ, were ǫ = + +4 σ is a generator of te Galois group Gal(L /k).. Preliminary Let L F (x) be te same as given in Section 1. Ten all four roots α,1,α,,α,3,α,4 of F (x) are as follows: +(+) +4± (+)( +4)+ (+) +4, 4 (+) +4± (+)( +4) (+) Let K = k( +4). Ten K is a unique quadratic subfield of L te fundamental unit ǫ of K is It is known tat L is a cyclic extension over k wit were σ is defined by Gal(L /k) = σ, Gal(L /K ) = σ, σ(α ) = (+ 1 + ) ( )α +( )α +(1 1 + )α3. We ave Lagrange resolvent r 1 = α,1 +α, i α,3 α,4 i for L, were i = 1, we find R 1 = r 4 1 = (+) ( +4)( i) F q (i)(t).

3 REGULATORS OF AN INFINITE FAMILY OF QUARTIC FUNCTION FIELDS 419 We notice tat te discriminant D K of K is +4 primes in k dividing + are ramified in L ; ence, according to conductor-discriminant formula, it follows tat te discriminant D L of L over k is given by D L = (+) ( +4) 3. Proposition.1. Te infinite prime of k splits completely in L ; so L as unit rank 3, L is a real function field. Proof. Let P be an infinite prime of k( 4 R 1 ) lying over k (resp. P ) be te infinite prime of k(i) (resp. k(i, 4 R 1 )) lying over (resp. P ). Ten we ave k(i) = F q (i)((t 1 )) k(i, 4 R 1 ) P = F q (i, 4 R 1 )((t 1 )). If we express R 1 = (+) ( +4)( i) in F q (i)((t 1 )), we ave R 1 = a 8 dt 8d + lower terms on t, were = d i=0 a it i for a i F q, (i = 0,1,...,d 1) a d F q. Tus we ave 4 R1 F q (i)((t 1 )) k(i, 4 R 1 = F ) P q (i, 4 R 1 )((t 1 )) = F q (i)((t 1 )) = k(i) ; tis implies tat k( 4 R 1 ) P = k, wic completes te proof. Te infinite prime of k splits completely in L ; so we ave k L k = F q ((t 1 )), were k is te completion of k at. For a nonzero element a = i= m c it i k wit m Z, c i F q (i m) c m 0, we define dega = m. Let U(L ) (resp. U(K )) be te unit group of te maximal order of L (resp. K ). Let U(L /K ) := { ǫ U(L ) N L /K (ǫ) = ǫ σ (ǫ) F q}. It is known [4] tat tere is η L wit U(L /K ) = F q η,σ(η ), we call η a relative fundamental unit of L over K. Let R(L ) (resp. R(K )) be te regulator of L (resp. te regulator of K ) for ǫ i (i = 1,,3) U(L ), R(ǫ 1,ǫ,ǫ 3 ) := det degǫ 1 degǫ degσ(ǫ 3 ) degσ(ǫ 1 ) degσ(ǫ ) degσ (ǫ 3 ). degσ (ǫ 1 ) degσ (ǫ ) degσ 3 (ǫ 3 ) Let D L /K (resp. D L /k) denote te discriminant of L over K (resp. L over k).

4 40 J. LEE AND Y. LEE 3. Determination of relative fundamental units In tis section, we sow tat te relative fundamental unit η of L over K is equal to a root α of F (x) = x 4 x 3 ( )x x+1 up to constant in F q. It is known [4] tat Q L := [U(L ) : U(K )U(L /K )] {1,} R(ǫ K,η,σ(η )) = Q L R(L ). We note tat for α U(L /K ) β U(K ), we ave ( R(β,α,σ(α)) = deg(β) (degα) +(degσ(α)) ). Tus, for determination ofr(l ) a relative unit η, we need a lowerbound an upper bound of (degη ) +deg(σ(η )). Proposition 3.1. Let η L be suc tat Ten we ave U(L /K ) = F q η,σ(η ). 4.5(deg) (degη ) +deg(σ(η )) 5(deg). Proof. Since α U(L /K ), we ave for integers a,b α = η a σ(η ) b ( (degα ) +(degσ(α )) = (a +b ) (degη ) +deg(σ(η )) ). We note tat Tus, we ave Finally, we obtain tat α = σ(α ) = degα = deg degσ(α ) = deg. (degη ) +deg(σ(η )) (degα ) +(degσ(α )) = 5(deg). Now, we note tat D L = N K /k(d L /K )D K = (+) ( +4) 3. Since D K = +4, we ave tat N K /k(d L /K ) = (+) ( +4).

5 REGULATORS OF AN INFINITE FAMILY OF QUARTIC FUNCTION FIELDS 41 Moreover, we ave Tus we ave D L /K (η σ (η )) N K /k(d L /K ) N K /k(η σ (η )). (+) ( +4) (η σ (η )) (σ(η ) σ 3 (η )). We observe tat for c 1,c F q, σ (η ) = c 1 /η, σ 3 (η ) = c /σ(η ) deg(η c 1 /η ) = degη, deg(σ(η ) c /σ(η )) = degσ(η ) ; tus, we ave deg (+) ( +4) degη + degσ(η ) ( (degη ) +(degσ(η )) )1, so tat we get ( ) deg (+) ( +4) 8 ( (degη ) +(degσ(η )) ). Consequently, we obtain tat 4.5(deg) (degη ) +(degσ(η )). Teorem 3.. A root α of F (x) is a relative fundamental unit of L over K up to constant in F q. Proof. Since α U(L /K ), we ave for integers a,b α = η a σ(η ) b ( (1) (degα ) +(degσ(α )) = (a +b ) (degη ) +deg(σ(η )) ). From (1) Proposition 3.1, it follows tat 5(deg) 4.5(a +b )(deg). Tus, we ave a +b = 1; tis implies tat α is η ±1 or σ(η ) ±1, wic completes te proof.

6 4 J. LEE AND Y. LEE 4. Proof of te main result In tis section, we first compute Q L, we ten complete te proof of Teorem 1.1. We need te following two lemmas. Lemma 4.1 is a criterion for determining weter Q L is 1 or not. A similar criterion in te number field case is given in [4]. Lemma 4.1. Let U(K ) = F q ǫ U(L /K ) = F q η,σ(η ). If cǫ η 1 σ is not a square in U(L ) for any c F q, ten Q L = 1. Proof. SupposetatQ L 1.Tenwecantakeu U(L ) U(K )U(L /K ). As u 1+σ U(K ), we ave u 1+σ = c 1 ǫ λ+1 If u 1+σ = c ǫ λ (c F u q ), ten ǫ λ U(L /K )U(L /K ). Tus we ave u 1+σ = c 1 ǫ λ+1 or u 1+σ = c ǫ λ (c 1,c F q ). ( ) σ F q wic implies tat u u ǫ λ c 1 ǫ = u ǫ λ ( u ǫ λ If we let u 1 = u U(L), ten we ave ǫ λ Since u 1+σ 1 U(L /K ), we ave ) σ. c 1 ǫ = u 1+σ 1 (c 1 F q). () u 1+σ 1 = c 3 η A σ(η ) B (c 3 F q A,B Z). (c 1 F q ); so we get We note tat if A B ave te same parity (tat is, bot are even or odd), ten ( ) ησ(η A ) B = c 4 η A+B 1+σ σ(η ) A+B (c 4 F q); terefore, we get (u 1 /(η A+B 1+σ σ(η ) A+B )) F q ; so we ave u 1 U(L /K )U(L /K ), wic is a contradiction. Tis sows tat A B ave not te same parity. In oter words, A 1 B ave same parity. Tus () implies tat (u 1 /(η A+B 1 1+σ σ(η ) A+B+1 )) = c3 η (c 3 F q ). Since (η A+B 1 σ(η ) A+B+1 ) 1+σ F q, by letting u := u 1, η A+B 1 σ(η ) A+B+1

7 REGULATORS OF AN INFINITE FAMILY OF QUARTIC FUNCTION FIELDS 43 we ave (3) u 1+σ = c 3 η u 1+σ = c 4 ǫ (c 3,c 4 F q); terefore, c 5 ǫ η 1 σ = u for u U(L ) c 5 F q, wic completes te proof. We first sow tat cǫ η 1 σ is not square in U(L m ) for any c F q. Ten by Lemma 4.1 we get Q L = 1. It ten follows tat R(L ) = 10(deg) 3 U(L ) = F q α,σ(α ),ǫ, were ǫ = It is tus enoug to sow tat cǫ η 1 σ is not square in U(L ) for any c F q. To determine weter cǫ η 1 σ is a square in U(L ) or not, we need te following lemma. Lemma 4.. Let E be a quadratic extension of F. If τ E is square in E, ten N E/F (τ), Tr E/F (τ) + N E/F (τ) Tr E/F (τ) N E/F (τ) are square in F. Proof. Using te following formulas in Proposition 3.1 in [15], te result follows immediately. τ + N E/F (τ) τ = Tr E/F (τ)+ N E/F (τ) τ N E/F (τ) τ = Tr E/F (τ), N E/F (τ) Proof of Teorem 1.1. In Teorem 3., we find tat η = cα (c F q ). Let τ = cǫ α /σ(α ) (c F q). We note tat N L /K (cτ ) = c ǫ, Tr L /K (τ )+ N L /K (τ ) = cǫ ( 3 1 ) (+) +4, ) Tr L /K (τ ) N L /K (τ ) = cǫ ( (+) +4. Let ( δ,1 := cǫ 3 1 ) (+) +4,

8 44 J. LEE AND Y. LEE ( δ, := cǫ ) (+) +4. Moreover,if δ,1 K is squarein K, ten N K /k(δ, ) Tr K /k(δ,1 )+ N K /k(δ,1 ) Tr K /k(δ,1 ) N K /k(δ,1 ) are square in k. We note tat N K /k(δ,1 ) = c ( ), wic is not square in k for any c F q. Moreover,if δ, K is squarein K, ten N K /k(δ, ) Tr K /k(δ, )+ N K /k(δ, ) Tr K /k(δ, ) N K /k(δ, ) are square in k. We note tat (4) N K /k(δ, ) = 4c 4. If 1 is not square in O k /q, ten (4) is not square in k for any c F q. On te oter, if 1 is square in O k /q wita = 1 in O k /q, ten we ave Tr K /k(δ, )+ N K /k(δ, ) = c(a 4 +4a 3 +(8a +4a) +8a ) Tr K /k(δ, ) N Km/k(δ, ) = c(a 4 +4a 3 +(8a 4a) +8a ); bot are not square in k for any c F q. Hence, from Lemma 4., it follows tat δ,1 δ, are not square in K τ is not square in L. Tis completes te proof. 5. Infinitely many family of quartic function fields In tis section, we sow tat tere are infinitely many primes q suc tat ((t) + 4)((t) + )(t) is square free in F q [t], were (t) is a given monic polynomial in Z[t]. Consequently, Teorem 1.1 olds 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 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, ten f(x) f (x) ave no common factors in K[x]; tus tey are relatively prime. For te converse, if we assume tat f(x) is not square free, ten f(x) f (x) ave some common factor in K[x]; so f(x) f (x) are not relatively prime in K[x]. Lemma 5.. (1) Let K be a field f(x) K[x] be a square free polynomial. Ten for g(x) K[x], if f(g(x)) is relatively prime to g (x), ten f(g(x)) is square free in K[x]. () For f(x), g(x) Z[x], if f(g(x)) is square free in Q[x], ten f(g(x)) F q [x] is square free for every prime q, were f (resp. ḡ) denotes te reduction of coefficients of f (resp. g) modulo q.

9 REGULATORS OF AN INFINITE FAMILY OF QUARTIC FUNCTION FIELDS 45 Proof. (1) It is sufficient to sow tat f(g(x)) f (g(x))g (x) are relatively prime by Lemma 5.1. Since f(x) is square free in K[x], f(x) f (x) are relatively prime in K[x]; so f(g(x)) f (g(x)) are also relatively prime in K[x]. Tus, if f(g(x)) g (x) are relatively prime by our assumption, ten f(g(x)) is square free in K[x]. () If f(g(x)) is square free in Q[x], f(g(x)) g (x) are relatively prime in Q[x] by Lemma 5.1; ence, tere exist 1 (x) (x) in Q[x] suc tat Tus we ave f(g(x)) 1 (x)+g (x) (x) = 1. f(ḡ(x)) 1 (x)+ḡ (x) (x) = 1; equivalently, f(ḡ(x)) ḡ (x) arerelativelyprime. It tus followstat f(g(x)) is square free in F q [x] for every prime q from te part (1). Teorem 5.3. Let (t) be of te type t k +c F q [t] wit (c +4)(c+)c F q. Ten ((t) +4)((t)+)(t) is square free in F q [t] for any power q of a prime except q = or q dividing k. Proof. From Lemma 5.(1), we obtain te result. References [1] W. E. H. Berwick, Algebraic number-fields wit two independent units, Proc. London Mat. Soc. 34 (193), [] J. W. S. Cassels, An Introduction to te Geometry of Numbers, Springer-Verlag, Berlin, [3] H. Darmon, Note on a polynomial of Emma Lemer, Mat. Comp. 56 (1991), no. 194, [4] M.-N. Gras, Table numérique du nombre de classes et des unités des extensions cycliques réelles de degré 4 de Q, Publ. Mat. Besançon, 1977/1978, fasc., pp. 1 6 & [5], Special units in real cyclic sextic fields, Mat. Comp. 48 (1987), no. 177, [6] H. Hasse, Aritmetisce Bestimmung von Grundeineit und Klassenzal in zykliscen kubiscen und biquadratiscen Zalkorpern, Matematisce Ablungen, B 3, Walter de Gruyter, Berlin (1975), , Originally publised, [7] Y. Kisi, A family of cyclic cubic polynomials wose roots are systems of fundamental units, J. Number Teory 10 (003), no. 1, [8] A. J. Lazarus, On te class number unit index of simplest quartic fields, Nagoya Mat. J. 11 (1991), [9] O. Lecaceux, Unités d une famille de corps cycliques réeles de degré 6 liés à la courbe modulaire X 1 (13), J. Number Teory 31 (1989), no. 1, [10], Unités d une famille de corps liés à la courbe X 1 (5), Ann.Inst. Fourier(Grenoble) 40 (1990), no., [11] J. Lee Y. Lee, Fundamental units regulators of quintic function fields, preprint. [1] Y. Lee, R. Sceidler, C. Yarris, Computation of te fundamental units te regulator of a cyclic cubic function field, Experiment. Mat. 1 (003), no., [13] E. Lemer, Connection between Gaussian periods cyclic units, Mat. Comp. 50 (1988), no. 18, [14] D. H. Lemer E. Lemer, Te Lemer project, Mat. Comp. 61 (1993), no. 03,

10 46 J. LEE AND Y. LEE [15] S. R. Louboutin, Hasse unit indices of diedral octic CM-fields, Mat. Nacr. 15 (000), [16], Te simplest quartic fields wit ideal class groups of exponents less tan or equal to, J. Mat. Soc. Japan 56 (004), no. 3, [17] R. Scoof L. C. Wasington, Quintic polynomials real cyclotomic fields wit large class numbers, Mat. Comp. 50 (1988), no. 18, [18] D. Sanks, Te simplest cubic fields, Mat. Comp. 8 (1974), [19] Y. Y. Sen, Unit groups class numbers of real cyclic octic fields, Trans. Amer.Mat. Soc. 36 (1991), no. 1, [0] A. Stein H. C. Williams, Some metods for evaluating te regulator of a real quadratic function field, Experiment. Mat. 8 (1999), no., [1] F. Taine, Jacobi sums new families of irreducible polynomials of Gaussian periods, Mat. Comp. 70 (001), no. 36, [] L. C. Wasington, A family of cyclic quartic fields arising from modular curve, Mat. Comp. 57 (1991), no. 196, [3] H. C. Williams C. R. Zarnke, Computer calculation of units in cubic fields, Proceedings of te Second Manitoba Confference on Numerical Matematics (Univ. Manitoba, Winnipeg, Man., 197), pp Congressus Numerantium, No. VII, Utilitas Mat., Winnipeg, Man., [4] Z. Zang Q. Yue, Fundamental units of real quadratic fields of odd class number, J. Number Teory 137 (014), Jungyun Lee Institute of Matematical Sciences Ewa Womans University Seoul , Korea address: lee9311@ewa.ac.kr Yoonjin Lee Department of Matematics Ewa Womans University Seoul , Korea address: yoonjinl@ewa.ac.kr