Korean J. Math. 20 (2012), No. 4, pp. 477 483 http://dx.doi.org/10.11568/kjm.2012.20.4.477 HILBERT l-class FIELD TOWERS OF IMAGINARY l-cyclic FUNCTION FIELDS Hwanyup Jung Abstract. In this paper we study the infiniteness of the Hilbe l-class field tower of imaginary l-cyclic function fields when l 5. 1. Introduction Let k = F q (T ), A = F q [T ] and = (1/T ). For a finite extension F of k, write O F for the integral closure of A in F and H F for the Hilbe class field of F with respect to O F ([4]). Let l be a prime number. Let F (l) 1 be the Hilbe l-class field of F (l) 0 = F, i.e., F (l) 1 is the maximal l-extension of F inside H F, and inductively, F (l) n+1 be the Hilbe l-class field of F n (l) for n 1. We obtain a sequence of fields F (l) 0 = F F (l) 1 F (l) n, which is called the Hilbe l-class field tower of F. Hilbe l-class field tower of F is infinite if F n (l) We say that the F (l) n+1 for each n 0. For any multiplicative abelian group A, let r l (A) = dim Fl (A/A l ) be the l-rank of A. Let Cl F and OF be the ideal class group and the group of Received November 2, 2012. Revised December 10, 2012. Accepted December 15, 2012. 2010 Mathematics Subject Classification: 11R58, 11R60, 11R18. Key words and phrases: Hilbe l-class field tower, imaginary l-cyclic function field. This work was suppoed by the National Research Foundation of Korea(NRF) grant funded by the Korea government(mest)(no.2010-0008139). c The Kangwon-Kyungki Mathematical Society, 2012. This is an Open Access aicle distributed under the terms of the Creative commons Attribution Non-Conercial License (http://creativecommons.org/licenses/bync/3.0/) which permits unrestricted non-commercial use, distribution and reproduction in any medium, provided the original work is properly cited.
478 H. Jung units of O F, respectively. In [6], Schoof proved that the Hilbe l-class field tower of F is infinite if r l (Cl F ) 2 + 2 r l (OF ) + 1. Assume that q is odd, and let l be a prime divisor of q 1. By an imaginary l-cyclic function field, we always mean a finite (geometric) cyclic extension F of degree l over k in which is ramified. In [2, 3], we studied the infiniteness of the Hilbe 2-class field tower of imaginary quadratic function fields and of the Hilbe 3-class field tower of imaginary cubic function fields. The aim of this paper is to study the infiniteness of the Hilbe l-class field tower of imaginary l-cyclic function fields when l 5. We give a several criterions for imaginary l-cyclic function fields to have infinite Hilbe l-class field tower with some examples. 2. Preliminaries 2.1. Rédei matrix and the invariant λ 2. Assume that q is odd, and let l be an odd prime divisor of q 1. Write P for the set of all monic irreducible polynomials in A. Fix a generator γ of F q. Any l-cyclic function field F can be written as F = k( l D), where D = ap r 1 with a {1, γ} and P i P, 1 r i l 1 for 1 i t. Then F = k( l D) is imaginary if and only if l deg D. Let σ be a generator of G = Gal(F/k). Then we have (2.1) l 1 r l (Cl F ) = λ i (F ), i=1 where λ i (F ) = dim Fl (Cl (1 σ)i 1 F /Cl (1 σ)i F ). By genus theory, λ 1 (F ) = t 1. Let η = γ q 1 l. For 1 i j t, let e ij F l be defined by η e ij = ( P i P j ) l. Let d i F l be defined by deg P i d i mod l for 1 i t. Let R F = (e ij) 1 i,j t be the t t matrix over F l, where the diagonal entries e ii are defined by the relation t i=1 r je ij = 0 or d i + t i=1 r je ij = 0 according as a = 1 or a = γ. Then we have Proposition 2.1. For an imaginary l-cyclic function field F over k, we have (2.2) λ 2 (F ) = { t 1 rank R F if a = 1, t rank R F if a = γ.
Hilbe l-class field towers of imaginary l-cyclic function fields 479 Proof. Let R F be the (t + 1) t matrix over F l obtained from R F by adding (d 1 d t ) in the last row. By [1, Corollary 3.8], we have λ 2 (F ) = t rank R F. Using the relation t i=1 r je ij = 0 or d i + t i=1 r je ij = 0 according as a = 1 or a = γ, it can be shown that rank R F = 1+rank R F if a = 1 and rank R F = rank R F if a = γ. Hence we get the result. 2.2. Some lemmas. Let E and K be finite (geometric) separable extensions of k such that E/K is a cyclic extension of degree l, where l is a prime number not dividing q. Write S (F ) for the set of all primes of F lying above. Let γ E/K be the number of prime ideals of O K that ramify in E and ρ E/K be the number of places p in S (K) that ramify or ine in E. It is known ([2, Proposition 2.1]) that the Hilbe l-class field tower of E is infinite if (2.3) γ E/K S (K) ρ E/K + 3 + 2 l S (K) + (1 l)ρ E/K + 1. For D A, write π(d) for the set of all monic irreducible divisors of D. Lemma 2.2. Assume that l 5 is a prime divisor of q 1. Let r = 2 if l = 5 or 7 and r = 1 if l 11. Let F = k( l D) be an imaginary l-cyclic function field over k. If there is a nonconstant monic polynomial D such that l deg D, π(d ) π(d) and ( D P 1 ) l = = ( D P r ) l = 1 for some P 1,..., P r π(d) \ π(d ), then F has infinite Hilbe l-class field tower. Proof. Put K = k( l D ). Then K is an l-cyclic extension of k in which, P 1,..., P r split completely. Let E = KF. Applying (2.3) on E/K with γ E/K rl and S (K) = ρ E/K = l, we see that E has infinite Hilbe l-class field tower. Since E F (l) 1, F also has infinite Hilbe l-class field tower. Lemma 2.3. Assume that l 5 is a prime divisor of q 1. Let F = k( l D) be an imaginary l-cyclic function field over k. If there are two distinct nonconstant monic polynomials D 1, D 2 such that l deg D i, π(d i ) π(d) for i = 1, 2 and ( D 1 ) P l = ( D 2 ) P l = 1 for some P π(d) \ (π(d 1 ) π(d 2 )), then F has infinite Hilbe l-class field tower. Proof. Put K = k( l D1, l D2 ). Then K is a bicyclic l 2 -extension of k in which, P split completely. Let E = KF. By applying (2.3) on E/K with γ E/K l 2 and S (K) = ρ E/K = l 2, we see that E has
480 H. Jung infinite Hilbe l-class field tower. Since E F (l) 1, F also has infinite Hilbe l-class field tower. 3. Hilbe l-class field tower of imaginary l-cyclic function field Let l 5 be a prime divisor of q 1. Let F = k( l D) be an imaginary l-cyclic extension of k, where D = ap r 1 with a {1, γ}, P i P and 1 r i l 1 for 1 i t and l deg D. Since OF = F q, i.e., r l (OF ) = 1, by Schoof s theorem, the Hilbe 3-class field tower of F is infinite if r l (Cl F ) 5. Since λ 1 (F ) = t 1, F has infinite Hilbe l-class field tower if t 6. Let ϑ F be 0 or 1 according as a = 1 or a = γ. Then, by (2.2), we have λ 1 (F ) + λ 2 (F ) = 2t 2 + ϑ F rank R F. Hence, for the case t = 4 or t = 5, we have the following theorem. Theorem 3.1. Let l be an odd prime divisor of q 1. Let F = k( l D) be an imaginary l-cyclic function field with D = ap r 1. Assume that t = 4 or 5. If rank R F 2t 7 + ϑ F, then F has infinite Hilbe l-class field tower. Example 3.2. Consider k = F 11 (T ) and l = 5. Then γ = 2 is a generator of F 11 and η = 4. Let P 1 = T, P 2 = T + 1, P 3 = T + η and P 4 = T + η 1. We have e 12 = e 34 = 0, e 13 = e 24 = 2, e 14 = e 23 = 3. Let F = k( 5 D) with D = γp 1 P 2 P 3 P 4. Then F is an imaginary 5-cyclic function field over k and the matrix R F is 0 0 2 3 0 0 3 2 2 3 0 0 3 2 0 0 whose rank is 2. Theorem 3.1. Then F has infinite Hilbe 5-class field tower by In the following we will give more simple criterions for the infiniteness of Hilbe l-class field tower of F by using Lemma 2.2 and Lemma 2.3. Theorem 3.3. Assume that l = 5 or 7. Let F = k( l D) be an imaginary l-cyclic extension of k with D = ap r 1. Then F has infinite Hilbe l-class field tower if one of following conditions holds: (1) t 4 and l deg P i for 1 i 3,
Hilbe l-class field towers of imaginary l-cyclic function fields 481 (2) t 3 and l deg P i, ( P i P 3 ) l = 1 for i = 1, 2. Proof. For (1), choose x, y, z, w F l such that (x, y) (0, 0), xe 14 + ye 24 = 0 and (z, w) (0, 0), ze 14 + we 34 = 0. Let D 1 = P1 x P y 2 and D 2 = P1 z P3 w. We have l deg D 1, l deg D 2 and ( D 1 P 4 ) l = ( D 2 P 4 ) l = 1. Hence, by Lemma 2.3, the Hilbe l-class field tower of F is infinite. (2) is an immediate consequence of Lemma 2.3. Let N(n, q) be the number of monic irreducible polynomials of degree n in A = F q [T ]. Then it satisfies the following one ([5, Corollary of Proposition 2.1]): (3.1) N(n, q) = 1 µ(d)q n d. n For α F q, let N(n, α, q) be the number of monic irreducible polynomials of degree n with constant term α in A = F q [T ]. Let D n = {r : r (q n 1), r (q m 1) for m < n}. For each r D n, let r = m r d r, where d r = gcd(r, qn 1 ). In [7], Yucas proved that N(n, α, q) satisfies q 1 the following formula: d n (3.2) N(n, α, q) = 1 nφ(f) where f is the order of α in F q. r Dn mr=f φ(r), Example 3.4. Consider k = F 11 (T ) and l = 5. By using (3.1), we can see that there are 32208 monic irreducible polynomials of degree 5 in A = F 11 [T ]. Choose three distinct monic irreducible polynomials P 1, P 2, P 3 of degree 5. Then F = k( 5 T P 1 P 2 P 3 ) is an imaginary 5-cyclic function field over k whose Hilbe 5-class field tower of F is infinite by Theorem 3.3, (1). Example 3.5. Consider k = F 29 (T ) and l = 7. For any α F 29, we have ( α ) T 7 = α 4. Using (3.2), it can be easily shown that N(7, 1, 29) = N(7, 1, 29) = 88009572. Let P 1 and P 2 be monic irreducible polynomials of degree 7 in A = F 11 [T ] with P 1 (0) = 1 and P 2 (0) = 1. Then ( P 1 ) T 7 = ( 1 ) T 7 = 1 and ( P 2 ) T 7 = ( 1) T 7 = 1. Hence F = k( 7 T P 1 P 2 ) is an imaginary 7-cyclic function field over k whose Hilbe 7-class field tower of F is infinite by Theorem 3.3, (2).
482 H. Jung Theorem 3.6. Assume that l 11. Let F = k( l D) be an imaginary l-cyclic extension of k with D = ap r 1. Then F has infinite Hilbe l-class field tower if one of following conditions holds: (1) t 3 and l deg P i for i = 1, 2, (2) t 2 and l deg P 1, ( P 1 P 2 ) l = 1. Proof. For (1), choose x, y F l such that (x, y) (0, 0), xe 13 +ye 23 = 0. Let D = P x 1 P y 2. Then D is a monic nonconstant polynomial with l deg D and ( D P 3 ) l = η xe 13+ye 23 = 1. Hence, by Lemma 2.2, the Hilbe l-class field tower of F is infinite. (2) is an immediate consequence of Lemma 2.2. Example 3.7. Consider k = F 23 (T ) and l = 11. By using (3.1), we can see that there are 86619068901264 monic irreducible polynomials of degree 11 in A = F 23 [T ]. Choose two distinct monic irreducible polynomials P 1, P 2 of degree 11. Then F = k( 11 T P 1 P 2 ) is an imaginary 11-cyclic function field over k whose Hilbe 11-class field tower of F is infinite by Theorem 3.6, (1). Example 3.8. Consider k = F 53 (T ) and l = 13. Using (3.2), it can be easily shown that N(13, 1, 53) = 38515860836695985496. Let P be a monic irreducible polynomial of degree 13 in A = F 53 [T ] with P (0) = 1. Then ( P T ) 13 = ( 1 T ) 13 = 1. Hence F = k( 13 T P ) is an imaginary 13-cyclic function field over k whose Hilbe 13-class field tower of F is infinite by Theorem 3.6, (2). References [1] S. Bae, S. Hu and H. Jung, The generalized Rédei matrix for function fields, Finite Fields Appl. 18 (2012), no. 4, 760 780. [2] H. Jung, Hilbe 2-class field towers of imaginary quadratic function fields submitted. [3], Hilbe 3-class field towers of imaginary cubic function fields, submitted. [4] M. Rosen, The Hilbe class field in function fields, Expo. Math. 5 (1987), no. 4, 365 378. [5], Number theory in function fields, Graduate Texts in Mathematics, 210. Springer-Verlag, New York, 2002. [6] R. Schoof, Algebraic curves over F 2 with many rational points, J. Number Theory 41 (1992), no. 1, 6 14. [7] J.L. Yucas, Irreducible polynomials over finite fields with prescribed trace/prescribed constant term, Finite Fields Appl. 12 (2006), 211 221.
Hilbe l-class field towers of imaginary l-cyclic function fields 483 Depament of Mathematics Education Chungbuk National University Cheongju 361-763, Korea E-mail: hyjung@chungbuk.ac.kr