BULL. AUSTRAL. MATH. SOC. VOL. 63 (2001) 11R60 [21-34] CLASS NUMBER OF (v,n,M) -EXTENSIONS OSAMA ALKAM AND MEHPARE BILHAN An analogue of cyclotomic number fields for function fields over thefinitefield F, was investigated by L. Carlitz in 1935 and has been studied recently by D. Hayes, M. Rosen, S. Galovich and others. For each nonzero polynomial M in F, [T], we denote by k(AM) the cyclotomic function field associated with M, where k = F,(T). Replacing T by 1/T hi k and considering the cyclotomic function field Fv that corresponds to (1/T)V+1 gets us an extension of k, denoted by Lv, which is the fixed field of Fv modulo FJ. We define a (v,n,M)-extension to be the composite N = knk(Am)Lv where kn is the constant field of degree n over k. In this paper we give analytic class number formulas for (v,n,M)-extensions when M has a nonzero constant term. 1. INTRODUCTION Let F , be the finite field with q = pr elements, where p is a prime number, and let k = ¥q(T) be the rational function field. To each nonzero polynomial M(T) in RT = FJT] one can associate a field extension k(Att), called the Af** cyclotomic function field. It has properties analogous to the classical number fields. Such extensions were investigated by Carlitz [2] and have been studied in recent years by Hayes, Rosen, Galovich, Goss and others. Hayes (in [4]) developed the theory of cyclotomic function fields in a modern language and constructed the mayiTpal Abelian extension of k. We shall briefly review the relevant portions of Carlitz' and Hayes' theory. Let fc be the algebraic closure of k and A; be its underlying additive group. The Frobenius automorphsim $ defined by $(u) = vfl and the multiplication map ft? defined by HT{T) = Tu are Fg-endomorphisms of F 1 ". The substitution of $ + /*r for T in every polynomial M{T) 6 RT introduces a ring homomorpbism from RT into Endf« ) which defines an RT -module action on k. The action of a polynomial M(T) 6 RT on u e k is denoted by uM and given by uM = M ($ + nt)(u). Received 27th January, 2000 Copyright Clearance Centre, Inc. Serial-fee code: 0004-9727/01 21 SA2.00+0.00. 22 O. Alkam and M. Bilhan [2] This action preserves the F,-algebra structure of k, since up = pu for 0 € F , . Carlitz and Hayes established the following results. (1) If degAf = d, then uM = £ [ M l u*\ where (2) (3) (4) \M is a polynomial in i=0 L * J L* . Ml is the leading RT of degree (d — i)g*. Moreover = M and coefficient of M. uM is a separable polynomial in u of degree cp. If A M denotes the set of roots of the polynomial uM in k then AM is an .Ry-submodule of k which is cyclic and isomorphic to RT/(M) . The field k(AM), which is obtained by adjoining the elements of AM to k, is a simple, Abelian extension of k with a Galois group isomorphic to (RT/(M))*. By $ ( M ) we denote the order of the group (RT/(M))*. If M ^ 0 then the infinite prime divisor P^ of Jk splits into $(M)/(q - 1) prime divisors of fc(A*f) with ramification index eoo = 9 — 1 and residue degree /oo = 1 • Because of the presence of constant fields and wild ramification of the infinite prime Poo, the above Mth cyclotomic function fields fc(Ajif) are not sufficient to generate the maximal Abelian extension of k. To remedy this difficulty, Hayes constructed the fields Fv by applying Carlitz' theory with the generator 1/T instead of T and ( l / r ) " + 1 instead of M and considered the fixed field Lv of Fv under F£. Then the maximal Abelian extension A of k appears as the composite EKTLOO , where E is the composite of all constant field extensions of k, KT is the composite of all cyclotomic function fields and Loo is the composite of all fields Lv. Thus we deduce an analogue of the KroneckerWeber Theorem for rational function fields: Every finite Abelian extension K of k is contained in a composite of the type N = fcnfc(AAf)Lv, where kn is a constant field extension of degree n, M is a nonzero polynomial in RT and v is a nonnegative integer. We call such extensions (v,n,M)-extensions. In [3], Galovich and Rosen gave an analytic class number formula for the field HAM) when M = Pa for some prime polynomial P € F,[T]. In this paper we give an analytic class number formula for (v, n, M) -extensions for any nonnegative integer v, positive integer n and any polynomial M in Fq[T] with a nonzero constant term. Let N = knk(AM)Lv be such an extension. Then since k C Lv and AM is a cyclic RT -module, say AM = (A), N = FqnLv(\). Hence the fields N and LV(X) have the same genus. Moreover, the class number of N is divisible by the class number of LV(X). We shall give explicit class number formulas for both LV(X) and N. We begin by studying the decomposition of the infinite prime divisor Poo of fc in LV(X). Let GL = Gal(Lv(X)/k). Then GL is isomorphic to the direct sum of GM = Gal(fc(A)/Jfc) £ (RT/{M)Y and Gv = Gal(Lv/Jfc) [4]. [3] Class number of («, n,M)-extensions If a € Gal(Lv(X)/Lv) then c^.toMX) .. «,.*<*) implies that erx = ap since € G ff^.,,.^ **- Notice 23 that = ^ . . . . i . = identity auto- morphsim. Moreover |Gal(L.(A)/L»)| = |Gj»f| = $ ( M ) . Hence GaI(L,(A)/£») a Consider the following diagrams of field extensions and prime divisors \/ K> k P with 51 being a prime divisor of LV(X) lying over the prime divisors 3 and I of the fields Lv andfc(A)respectively, and P being a prime divisior offclying under both 3 and £. Restricting automorphisms in Gal(Lv(A)/I>,,) to k(X) makes an isomorphism between the decomposition groups D(UK/3) and D(£/P). It is an isomorphism between the intertia groups 7(91/3) and I(l/P) as well. Thus e(£/P) and /(9t/3) equal f{l/P). Therefore we can easily see the following. PROPOSITION 1 . Let 9t be a prime divisor of LV(X) lying over the infinite prime divisor P^ of k. Then (i) (ii) (iii) (iv) Since the only finite prime divisors of k that ramify infc(A)are the divisors of M and no finite prime divisor of k ramifies in Lv, the only prime diviors of k that ramify in £V(A) are the prime polynomials that divide M. 2. ANALYTIC CLASS NUMBER FORMULAS In this section we develop class number formulas for the fields LV(X) and N by studying their L-functions and zeta functions. For the rest of this section the constant term of the polynomial M is assumed to be nonzero. . Let x be a character of GL = Gal(i u (A)/Jk). Then the L-functions are given by T H E FIELD LV(X) of Lv(X)/k L(s,x,LvW/k)=l[(l-W) V , R^) 24 O. Alkam and M. Bilhan [4] where <p runs over all prime divsors of k, and \ where P runs over all finite prime divisors of A;. Thus -l If X ^ Xo is a character in GL then L'{s,X,Lv{\)/k)= <?eF,[r],prime By x(Q) we mean the value of the character x at the Frobenius substitution of Lv(X)/k at Q. Therefore Hence L*(s,X,Lv(\)/k)= NAs where A = Since NA = qde&A for each monic polynomial A in F,[T], we can write L* (s, X, Lv{X)/k) = f; £ § £ , »=o where Si(x) = E Re(s) > i Class number of (v, n, M) -extensions [5] THEOREM 1 . Let M be a polynomial m d e g M = m ^ 1 and x^Xo PROOF: 25 in F 4 [T] with a nonzero constant GL then Si(x) = 0 for all i^m term. If + v + 2. Let t ^ m + v + 2 and St = { (A + (Af), A + <(1/T)"+1>) : A € Fq[T), monic of degree i with (A,M) = l } . Define 6 : St -» GL = Gal(Lv(\)/k) to be the map which sends ( A + ( M ) , 3 + <(l/r) v + 1 >) to (RA + (M), A + <(l/T)" +1 )) where RA is the unique polynomial in F,[T] such that A = M*QA + RA, degB^ < degM. Clearly 8 is well-defined. We show that 0 is onto. » m i=o j=o Suppose that R = £ TjTj (with Tj = 0 when j > degR), M = 53 dj-T', and /i = 53 a °i> = 1 a11^ allowing to have some of the Oj's to equal zero. j{l/T)V~3 Then, with the convention that Tj = 0 for all j such that degiZ < j < v, when deg.R < v the system do di 0 do d2 di 0 do dv-i dg-2 .d,, 0 .. . .. • .. ••• 0 0 0 X2 do .Xv. - rv . has a unique solution since the constant term a^ of M is nonzero. Let xo = qo, x\ = qi, ,xv = qv be the solution of that system and consider Q = 9u+2>--- ,9i-m-i chosen arbitrarily and qi-m = d^1. (Thus we have g*-"*-"-2 distinct choices for Q.) Take A = M*Q + R. Then since (i2,Af) = 1, we have (A, M) = 1. Moreover A is monic, deg A = i and This shows that © is onto. Now each g & GL corresponds to g»- m -»- 2 distinct choices of A. Moreover, if Ai = M*Qi + Ru A2 = M*Q2 + R2 then 26 O. Alkam and M. Bilhan [6] Therefore jleF,[T],monic (AAf) = 0. This completes the proof of the theorem. D The previous Theorem tells us that the ^-function L*{s,x,Lv(X)/k) is a polynomial in q~s with degree at most m + t) + l whenever x # Xo- We may consider i^ to be a subgroup of Gal(fc(A)/fc) via identifying each o € ^ J with aa € Gal(A;(A)/A;) which maps A to aX. If we let 5 = {(<ra,T) : a € FJ, r € Gv = Gal(Lv/k)} then 5 is a subgroup of GL = Ga\.(Lv(X)/k). Moreover, |5| = (g — l)qv. The subgroup 5 is the decomposition group of the point at infinity. 1: A character x of Gal(*(A)/ft) is said to be real if x(o) = 1 for all a € F£, while a character x of Gal(iv(A)//:) is said to be real if *(«) = 1 for all s € 5 . Clearly there are ($(M)/(g — 1)) — 1 nontrivial real characters of each Galois group. Moreover, for any nontrivial real character % of Gal(fc(A)/fc), £*(0, x> K^)/k) = ° [31DEFINITION THEOREM 2 . For any nontrivial real character x L*(0,X,Lv(\)/k)=0. of Gal(Lv(X)/k), PROOF: Any nontrivial real character x of Gal(Xv(A)/fc) can be viewed as a character of Gal(fc(A)/fc) via defining x(ff) = X ( * , 1 G . ) . Moreover, L*{s,x,Lv{X)/k) = L* (s, x, k(\)/k). Hence L* (0, x, Lv(X)/k) = 0 and the Theorem is proved. D In light of the previous results, we may proceed to derive a class number formula for the field ^ ( A ) . By Theorem 1 and Proposition 1 we may write the zeta function of Lv{\) as follows L'(s,X,Lv(X)/k) -<? l - s r 1 n L-(s,x,Lv It is well known that [7] Class number of («, n, Af) -extensions 27 where F(g~ 5 ,X,,(A)) is a polynomial in Z>[q~*] of degree 2g (where g is the genus of Lv(\)). Moreover, the class number of Lv(\) is F(l,Lv(\)) [5]. Thus ll m x^xo -( n ?J* )( n -( n £ = r i :^)( n "f xi XnonreaJ By Theorem 2, Z,*(0,x,L,,(A)/fc) = 0 for each nontrivial character x m GL- Using L'Hopital's rule to evaluate the limit of the above equation's right-hand side as 5 tends to 0, we derive the following class number formula: n»+ti+l n X€GL,ieal L x i m+v+l E -iSi(x))[ n i = 1 X 6G t ,no X#X0 Lv{X)¥qn . Let GN = Gal (N/k), Gv = Gal (Lv/k) and G w = Gal(ik(A)/A). Then Gff essentially equals the direct sum of the groups GM, GV and the cyclic group Z n [4]. We shall study the ^-functions i*(s,x, N/k) for any nontrivial character x of Let x # Xo be a character in GAT • Then we have one of two cases: T H E FIELD I. The restriction of x t o GM ®GV = Gal(Lv(A)//:) is the trivial character. In this case we define the character ^ on Gal(fcF,n) by *(a) = x{(^GM, 1 G « , O ) ) We identify the restriction of x t o GM © Gv with the character Xres of G*f © G v which is defined by Xres ((f, T)) = X((<T,T,0)). Notice that x(((r,T,a)) — *(°) f° r each (CT, T, O) € Gff and that 'd' is nontrivial since Xres is the trivial character. Moreover, ^ can be viewed as a character of GN via putting \p((0-,T,a)) = \P(a). Hence L'(s,V,N/k) = L'(s,9,k¥qn/k). That is, Lm(s,X,N/k) = L'(s,*,kFqn/k). Thus our problem of studying L*(s,x,N/k) is reduced to studying L*(s,^,kWqn/k) which de equals £ * ( / ) / y s / , R e ( s ) > l , where (see [1]) CASE ( mod ») 28 O. Alkam and M. Bilhan [8] Let Tdf be the unique integer such that d e g / = c*n + r ^ , 0 ^ r<tf < n. Then and L'{s,%kFqn/k)= /£F,[T],monic where d/ = deg / . oo We can write L*(s,V,k¥qn/k) as £ Si(*)/?", »=o o Re(s) > 1, where Si{V) Since we have g* possible monic polynomials in F,[T] of degree i, Therefore L , t=0 Re(s) * i=0 i=0 oo »=0 * 1 Whence, if x is a nontrivial character of GN which is trivial on GM © Gv and is the character of Z n defined by ^fx(i) = x((lGAf, l<3oii)) then II. The restriction of x to Gjif © Gv is not the trivial character. Again we let Xres be the restriction of x t o GM © <?„, that is, Xres((o", T)) x ((<r,T,0)). Then CASE L*(s,X,N/k)= Yl x(( — <>, 3 ( ( / ) ) , d JAL )i Z) ^- ' Re(5)>l, [9] Class number of (v, n,M)-extensions where dA = degyl, ~A = A/TdA 29 and raA is the unique integer such that d* = c*n + UA, 0^rdA <n, [1]. If Ae*q[T],monic {A,M)=1, dA=i then f^^-, Re(5)>l. t=0 For each i, SiU) = x((lGM,lGv,ri))X(U+ (M), A+ {(^) V+1 ), o)Y £ (^,AT)=1, dA=i Since x((l<3M'^<'v' r *)) ^ independent of the choice of A as long as degyl = t, we have Si{x) = x((lGM,lGv,Ti)) J2 x(U+(M),A+ ^€F,[T]^nonic ( ( | ) V + 1 ) , o)) = 0 VV 7 / because Xres is nontrivial on GM © Gv. Therefore 5»(x) = 0 for all i > d.M + v + 2. Whence i=o To summarise we write { - f f (1) >—' Z »=o 2tW, 9 tf X r e s fa t r i v l otherwise. 2: A character x of GN = GaL(N/k) "a, T, m)) = 1 for any a € f j , T 6 G« and m € Z n . DEFINITION is said to be real in GN if Clearly we have ($(Af)/(g — 1)) — 1 nontrivial real characters in GAT . THEOREM 3 . Let x be a nontrivial real character in Gjv- Tien L*(0,x,N/k) = 0. PROOF: The character Xres is a nontrivial real character of GM © <?„. Hence IS{s,X,N/k) = t=0 30 O. Alkam and M. Bilhan [10] where AeF«fT],inonic (A,Af)=l, dA=i Since x is real, x{0-GM, lGv,ri)) = 1. Thus S*(x) = $(*>«,)• Therefore The Theorem then follows from Theorem 2. = L* (s, Xxes, Lv(\)/k). L'(s,x,N/k) D Having studied the Zr-functions L* (s, x> N/k), one can give a class number formula for N via exploring the zeta function £(s,N). Let I be a prime divisor of N lying over the infinite prime divisor PQQ of k and let p be a prime divisor of Lv(\) lying under £ and over P^. Then we deduce (from the theory of constant field extensions) that g(£,p) = (rfic(A)(p)»n) = (l,n) = 1. Thus, every prime divisor of LV(X) which lies over the infinite prime divisor of k has a unique extension to a prime divisor of N. Moreover, as is well known from the theory of constant field extensions, no prime divisor of LV(X) is ramified in N. Thus e{£/p) = 1. Hence f(£/p) = n . Therefore N£ = NpWM = qn. So Since the field of constants of N is ¥qn we get where F(q~ns,N) € Z[g" n s ] and F ( l , AT) = fc(iV); the class number of N. Thus L'(s,X,N/k) - , - ) - ( JI x/ xw Xres nontrivial xres trivial [11] Class number of (r,n, M)-extensions •( n 31 - X#XO Xres nontrivial X#X0 Xres nontrivial where wo, wi, . . . , un-\ are the nth roots of unity, w -( n ^::^ ')( G .n l G N Xres nontrivial X = 0 for all nontrivial real characters x € GN- If we By Theorem 3, L*(0,x,N/k) evaluate the limit of the right hand-side as 5 tends to 0 we get the following formula for the class number h(N): dM+v+l v . =( n '- E -«ftw)( ^n dM+v+l )( X eG w ,real *~1 x6GAr,nonreal xA Xres nontrivial w X#XO * * 0 3. EXAMPLES When we specialise our results to N = WqnLv(\) with n = 1 and v = 0 we get and n where m = degM and 5j(x) = 53 *(a + AgF 4 [7],monic d 4 i That is exactly the result obtained by Galovich and Rosen [3]. In the foDowing examples we apply the class number formula mentioned above for the special cases when F , = Z 2 , F, = Z 3 and for specific prime polynomials M(T) € ¥q[T]. 32 O. Alkam and M. Bilhan [12] E X A M P L E 1. Let k = Z2(T) and M(T) = T3+T+l. Then [JV : k] = $ ( M ) = 2 3 - 1 = 7. Thus GN = (Z 2 [T]/(T 3 + T + 1))* is cyclic of order 7. Hence the character group GN is cyclic of the same order. The element [T] in (Z 2 [T]/(r 3 + T + l » * could be identified with a generator for GN • Let x be a generator for the group GN and assume that x{[T]) = C> ^ e n C is a primitive 7 t n root of unity. Since FJ = Z$ = (1), any character of GN is real. Moreover S^tp) = S3(if>) = 0 for each ip € GN. Therefore Now and S2(xn) = xn{[if) + xn{[T]5) The number £ could be any primitive 7" 1 root of unity, in particular e2*"*/7. Substituting this value of £ in the class number formula yields h(N) = 71. 2. In this example we consider k = Z 3 (r) and M(T) = T2 + 1. Clearly GN = (Z 2 [T]/(T 2 +1))* is cyclic of order $(Af) = 3 2 - 1 = 8. The element [T + 1] is a generator for GN • Let x be a generator for GN • Then x([^ + 1]) is a primitive 8 t n root of unity, let us say x([^ + 1]) = C = e""i/4. A character x n is real if and only if n e {0,2,4,6}. Therefore EXAMPLE =0 m ro i=0 If we compute 5j(x ) we find that 52(x ) = 5 3 (x m ) = 0 for any m such that 1 ^ m ^ 7, and that 5o(x m )= B62s[T],moni • c degB=0 [13] Class number of (v,n,M)-extensions 33 Thus So(xm) = 0 when m is odd. Similarly we find that 5i(x m ) = C6"1 + C" + C 7m = e3l?Mr</2 + emiti^ + e~mvi/4. Substitution of these values in the class number formula gives that h(N) = 9. Having treated very special cases in the examples above, one may wonder about the more general case when F 9 = Zp and M(T) is any prime polynomial in ZP[T]. Let k = ZP(T) and let M(T) be any prime polynomial in ZP[T] of degree d. The extension k(Ani)/k is of degree $ ( M ) = jpd — 1 and the Galois group G = Gal(fc(AM)A) is isomorphic to (ZP[T]/M(T))* which is cyclic. We identify a generator of G with a generator [A] of (ZP[T]/M(T)) *. The character group G is cyclic as well. Moreover, if x # Xo is a generator of G then x([-^]) is a primitive (pd - l)st root of unity, say x([A]) = C = e2vi/(pd~^. Let H be the subgroup of G consisting of all real characters, that is H =iip e G : *([o]) = 1 for each a e Z * | , GENERAL TREATMENT. then \H\ = \G\/\ZP\ = (p* - l ) / ( p - 1) and H is cyclic generated by xp~x- Thus H = {x" 1 ^- 1 ) : 0 ^ m^pt/ip-i)}. If ft = {1,2, . . . , p * - 2 } and /*„ = { m ( p - 1) | d 1 ^ m ^ (p — l)/(p — 1) — l } , then a nontrivial character ij> is real if and only if ip = Xn f° r some n€hd- The class number h(k(AM)) of the field fc(Ajif) is given by d+l d+l f V // V d+1 d+1 £) ( (n E A =(n £,-&&)) where AM = B£Zp[T],monic Since G is cyclic, for any B € ZP[T] of degree i with 0 ^ t ^ d - 1 there is a unique nonnegative integer n [B ] with 0 ^ n [B ] ^ Pd - 1 such that [B] = ([.4])"1-81. Thus, Si(xm) = B£Zp[T],monic deg B=i BeZp[T],monic degB=i 34 O. Alkam and M. Bilhan [14] Hence n -c E -is(xn^)) (n E :-• E degB=t (nf v E [ »=0 B€Zp[T],monic ngft d »=0 BgZdeg p[T],ii B=i Replacing C by eiT%f\p ~ 1 /,n[B]'s by their values and evaluating the expression above gets us the sought class number. REFERENCES [1] M. Bilhan, 'Arithmetic progressions of polynomials over a finite field', in Number theory and its applications (Ankara 1996), Lecture Notes in Pure and Applied Mathematics 204 (Dekker, New York, 1999), pp. 1-21. [2] L. Carlitz, 'On certain functions connected with polynomials in a Galois field', Duke Math J. 1 (1935), 137-168. [3] S. Galovich and M. Rosen, 'The class number of cyclotomic function fields', J. Number Theory 13 (1981), 363-375. [4] D.R. Hayes, 'Explicit class field theory for rational functional fields', Trans. Amer. Math. Soc. 189 (1974), 77-91. [5] A. Weil, Basic number theory (Springer-Verlag, Berlin, Heidelberg, New York, 1973). Department of Mathematics College of Science University of Petra Amman Jordan Department of Mathematics Faculty of Science and Arts Middle East Technical University Ankara Turkey