Kuroda s class number relation

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1 ACTA ARITMETICA XXXV (1979) Kurodas class number relaton by C. D. WALTER (Dubln) Kurodas class number relaton [5] may be derved easly from that of Brauer [2] by elmnatng a certan module of unts, but the technque s applcable to a much wder class of relatons whch are obtaned from norm relatons. The man am here s to treat the case n whch several radcals of the same prme degree are adjoned to the ratonal feld. 1. Norm relatons. Let be the alos group of a normal extenson K/k of algebrac number felds and the sum of the elements n a subgroup. Then a relaton of the form (1.1) b( ) = 0 (b() «) s called a norm relaton. These have been studed by Rehm n [7] and are so-called because Artn has establshed n [1] that the relaton holds precsely when ( N K / K x) ) b( ) ( = 1 for all x K*. ere K s the subfeld fxed by and N s the relatve norm. If 1 denotes the character on nduced by the unt character on then the equaton (1.2) 1 (g) g = gg g g may be used to convert the norm relaton (1.1) nto the character relaton (1.3) b( ) 1 = 0. The most nterestng relatons satsfy two further condtons : (1.4) DEFINITION. b() = 0 s called a drect norm relaton f () there s an 0 S = { b() 0} such that 0 for all S, and () dstnct 1, 2 S 0 = { S 0, } satsfy =. Work completed under a Rouse Ball studentshp from Trnty College, Cambrdge.

2 42 C. D. Walter Ths defnton and ts notaton wll be subsumed from here on. All sums and products wll extend over S and wll ndcate ther restrctons to S 0. For any left (resp. rght) []-module M let M (resp. M) be the submodule fxed under the acton of and wrte M 0 for M. If M s torson-free over and M = 0 then (1.5) M 0 = M s a drect sum. For suppose, S 0 are dstnct. Then and so provdes / representatves for each coset of n. Thus acts a multple of the trace on M/M. Consequently f m = m M 0 wth m M then m = m because M = 0. ence, m s unque as M s torson-free. (1.6) TEOREM. A drect norm relaton has the form / / = / / ( ) and ts assocated character relaton s Moreover ( ) 1 = = { S 0 }, = { S 0 }, and S 0 completely specfes the relaton. Proof. When a ( ) = 0 s multpled by / for S 0 or = one obtans ( a 0 + a( )) + a( ) = 0 where a 0 = a( 0 ) and the sum extends over n S 0 {}. Ths gves a 0 + a() = 0 for S 0 and a () = 0 for =. Thus the form of the norm relaton s establshed. (1.3) gves the character equaton whch wll henceforth usually be wrtten (1.3 ) a( ) 1 = 0. (1.7) EXAMPLE. If s an elementary abelan group of prme exponent p and order p n and T s the set of (p n )/(p ) maxmal subgroups then = (p n )/(p ) + p n 1 T s a drect norm relaton.

3 Kurodas class number relaton 43 Proof. Any somorphsm between and ts character group * provdes a bjecton between maxmal and mnmal subgroups, namely = {g h(g) = 0 h *} where * s the mage of n *. The order of T s the number (p n )/(p) of mnmal subgroups. Now g f and only f g. So the number of maxmal contanng g s the number of mnmal subgroups of g, namely (p n )/(p) f g 1. Thus the norm relaton holds. It s drect because dstnct maxmal subgroups and satsfy = p n 2. There are several ways of constructng new relatons from gven ones by passng from the whole group to a subgroup or quotent group and vce versa (see [8]). In partcular, (1.8) LEMMA. Suppose and are subgroups of 0 such that = 0 and b( ) = 0 s a drect norm relaton for. If = {hg h, g } s a subgroup of 0 for every S then b() = 0 s a drect norm relaton for 0. Ths s clear because = = for S. 2. Brauers class number relaton. Let U be the unt group of K ; W ts subgroup of roots of unty; w 2 () the 2-component n the order of W; h() the class number of K; r() the rank of U/W ; and n() the degree of K/k. A bar wll denote the natural map U U/W. Choose one prme dvsor n K of each nfnte prme n k and suppose {C 1 r} s the set of ther decomposton groups n K/k. So r = r() + 1 and each C s determned up to conjugacy. If L s defned by the exact sequence r (2.1) 0 of []-modules where n õ n formulated as follows. []C L 0 = 1 then Brauers theorem may be (2.2) TEOREM ([9], Theorem 4.1). Suppose a( ) 1 = 0. If the submodule M of U s []-somorphc to L then a( ) a( ) h ( ) = ( n ( ) w2 ( )[ U : M ]). Unt groups may be wrtten n ether addtve or multplcatve notaton but the context wll clarfy the choce. Suppose (2.3) «U = {ε U n, n 0, wth nε U }

4 44 C. D. Walter s the group of unts wth powers n k. Then V = V QU for any subset V U and so the equaltes hold n the defntons below. Q* = [ 0 L : L 0 ], Q = [ 0 U : 0 W+U 0 ] = U : ], [ 0 U0 Q 0 = [ 0 U : ( 0 U «U) + U 0 ] = U : U + ], [ 0 0 U0 I() = [U «U : W+U] = [ U : U ], I 0 = [U 0 «U : W 0 + U ] = [ U 0 : U ]. By comparng ranks I() and I 0 are fnte. If x 0 X for some []- module X then (1.1) gves b( 0 )x = b() /0 x + b() /0 x As /0 s the trace for 0 X/X so (1.6) shows that [: 0 ]x X 0. Thus (2.4) [ 0 X : X 0 ] s fnte f X s fntely generated, and all the ndces above are fnte. The basc smplfcaton of (2.2) for drect norm relatons s : (2.5) LEMMA. [ a( ) a ( ) U : M ] = ( 0 / *) 0 a Q I ( ) Q. Proof. ( 0 U + U0) / U0 0 U / U0 whence (2.6) Q/Q 0 = I( 0 )/I 0. Let V = U0. Then Q *[ 0 U : 0M ]/[ U : M ] I ( 0) Q0 = [ 0M : M0][ 0U : 0M ][ M : U ][ U : V ][ U0 : 0U ] = U : M ][ M : ] = U : M + ] snce (M 0 +V)/M 0 V/M [ 0 0 V [ 0 0 V = [ U 0 / V : ( M0 + V ) / V ] = [( U + V ) / V : ( M + V ) / V ] by (1.5) = [ U : ( M + V ) U ] = [ U : M ]/[ M + U : M ] = [ U : M ]/[ U : M U ] = [ U : M ]/[ U : M ] I( ). Theorem (1.6) completes the proof.

5 Kurodas class number relaton The ndex Q*. Let L be defned to make the []-module sequence r (3.1) 0 []C L 0 = 1 exact. Assocated wth t s the submodule L 0 = L and the ndex Q * = [ 0 L :L 0 ] whch s fnte by (2.4). Both (2.1) and (3.1) are exact when restrcted to the submodules fxed by a subgroup because ths s a left exact functor and any pre-mage of an element n L or L s certanly fxed by. ence and so 0 L/L 0 { 0 ( (3.2) Q* = []C )}/{ ( []C )} ( 0 []C / ( []C )) Q *. 0 L /L 0 Now defne a parng on «L «L by (x, y) = (11 1)( xy*) where * s the nvoluton nduced by g õ g for g. If N s a -submodule of L wth bass {n r } let R(N) = det((n r,n s )) be the regulator of N. Ths s ndependent of the choce of bass and for another submodule N t satsfes (3.3) R(N) = [ N : N ] 2 R(N) whenever [ N : N ] s defned. Let gc denote the sum of the dstnct elements n {hgc h, c C }, gc the number of such elements, and gc ts mage n L under (3.1). If and S 0 are dstnct then there are h and h such that hh = g for any gven g. So h gh = 1 and g =. Thus (gc )(g C )* and gc, g C ) = 0. ( owever, the gc form a bass of L 0 for S 0 and sutable g because L 0 = L s a drect sum by (1.5). ence the correspondng matrx for R(L 0 ) s zero except for blocks of determnant R(L ) on the dagonal. Ths gves (3.4) R(L 0 ) = R(L ). The number of gc whch have elements s {g gγ g } / = 1 (γ ) where γ generates C. Thus the number wth 2 elements s r 2 () = 1 (1 γ )/2. Set r () = dm []C. As (3.1) s exact when fxed

6 46 C. D. Walter by so { gc } s a bass of L when g runs over representatves of the non-prncpal double cosets \/C. If gc hc then and ence ( gc, hc ) = gc hc / ( gc, gc ) = gc gc 2 /. R(L ) = r ()+1 2 r2() 1C det A where A = I ( gc / ) g,h for the dentty matrx I. Add together the rows of A to obtan the constant row 1C / and use t to elmnate ( gc / ) g,h. Thus det A = 1C / and (3.5) R(L ) = r ()+1 2 r2() /. Equaton (3.3) gves Q * 2 = R(L 0 )/R( 0 L ) and combnng ths wth (3.4) and (3.5) produces (3.6) Q * 2 = ( r ()+1 / ) / ( 0 r (0)+1 / ) because r 2 () = 1 (1 γ )/2 = power of 2. Now 1 (1 γ )/2 = r 2 ( 0 ) removes the 0 r() + 1 = dm L + 1 = dm []C = Thus (3.6), (3.2), and (1.6) together yeld (3.7) Q* 2a 0 = a()(r()+1). (r () + 1). 4. The Enhetenndex I(). I(), whch wll be wrtten I(K/k) n ths secton, s a generalzaton of asses Enhetenndex ([3], 20) for an abelan extenson of «over ts maxmal real subfeld. Let k 2 k 1 k 0 be a tower of felds. The basc property s (4.1) TEOREM. I(k 2 /k 0 ) dvdes [k 2 :k 0 ]. Ths s clear from the next lemma because I(k 2 /k 0 ) dvdes I(k 2 /k 1 ) I(k 1 /k 0 ). (4.2) LEMMA. If k 1 /k 0 has no ntermedate felds and [k 1 :k 0 ] = p then I(k 1 /k 0 ) = 1 or p. In the latter case p s prme and k 1 = k 0 (ε) for some unt ε such that ε p k 0. Conversely, f p s prme and k 1 k 0 ( 1 ) has ths form then I(k 1 /k 0 ) = 1 or p accordng to whether or not k 1 s the unque extenson of k 0 wth the form k 1 = k 0 (ω) where ω p k 0 s a root of unty wth p-power order. Proof. Let U and W be the groups of unts and roots of unty n k, W p the p-sylow subgroup of W, and V 1 the subgroup of unts n k 1 wth some power n V 0 = U 0 W 1. Then I(k 1 /k 0 ) = [V 1 : V 0 ]. The norm N for k 1 /k 0

7 Kurodas class number relaton 47 nduces the pth power map on V 1 /W 1 and maps V 1 nto U 0. ence V 1 / V 0 has exponent p. Assume V 1 V 0. If ε V 1 V 0 then there s an m such that ε m U 0 but ε mp U 0. So k 1 = k 0 (ε m ) and p s prme. Moreover, f s ζ a prmtve pth root of unty and k = k (ζ) then k 1 /k 0 s cyclc wth generatng automorphsm α, say. The norm N extends to k 1 /k 0. Let q = [W 1 : W 1p ]. Then ε q(1 α) W 1p ζ for ε V 1. Thus f ε 1, ε 2 V 1 then a, b can be chosen such that p a or p b, and (ε 1 a ε 2 b ) q(1 α) = 1. So ε 1 aq ε 2 bq U 0 and ε 1 a ε 2 b V 0. ence ε 1 V 0 mples p b and ε 2 V 0 ε 1. Therefore V 1 /V 0 s cyclc of order p. Suppose k 1 k 0 ( 1 ) but k 1 = k 0 (ε) where ε p U 0. Then ω W 1p ζ and Nω = 1 gve ω ζ. So puttng ω = ε q(1 α) 1 for ε 1 V 1 yelds V q(1 α) 1 ζ. In fact, ε gves V q(1 α) 1 = ζ. The last part of the lemma holds because n ths case V q(1 α) 0 = 1 f and only f W 1p = W 0p. 5. The ndces Q 0 and Q. (5.1) LEMMA. Q 0 dvdes [ : 0 ] r() r(). Proof. Q 0 s the order of 0 U/(U U «U) ( 0 U+ «U)/( U 0 +«U) ϕ( 0 U)/ϕ(U 0 ) where ϕ: U U/«U s the natural map. For ε 0 U the norm equaton (1.6) gves so that [: 0 ]ε = (l S 0 ) /0 ε + [:] /0 ε (5.2) [: 0 ]ϕ(u) = [: 0 ]ϕ(u 0 ) [: 0 ]ϕ( 0 U) [:]ϕ(u) because ϕ(u) = 0. Snce the sums are drect by (1.5) and each ϕ(u) s torson-free, Q 0 dvdes the ndex [: 0 ] dmϕ(u) between the end modules. Fnally dm ϕ(u) = r() r(). (5.3) LEMMA. If [: 0 ] = n s the same for all S 0 then Q dvdes In r(0) r() for each S 0 where I = [ 0 U «U : 0 W + U 0 «U]. I dvdes I( 0 K/K) whch dvdes n. Proof. Let ϕ : U U/«U be the natural map. Then (5.2) yelds nϕ(u) = nϕ(u 0 ) nϕ( 0 U) ϕ(u).

8 48 C. D. Walter The sums are drect. ence Q = [ 0 U + «U : U 0 +«U] dvdes the ndex n dm ϕ(u) between the end modules. As and dm ϕ(u) = dm ϕ(u) = r() r() for S 0,, so Q dvdes n r( 0) r(). Now Q = [ 0 U : 0 W + U 0 ] (r() r()) = r( 0 ) r() = [ 0 U : U 0 +( 0 U «U)][U 0 +( 0 U «U): 0 W+U 0 ] = QI. Thus the proof s completed by (4.1). 6. Kurodas relaton. (6.1) MAIN TEOREM. Suppose the subgroups of the alos group of a normal extenson K/k of number felds satsfy a drect norm relaton ((1.4) and (1.6)) whose correspondng character relaton s (1.3). Then the class numbers h() of the felds K fxed by the subgroups are related by (6.2) h() a() = (w Q 0 ) a0 {I() [ : 0 ] (r())/2 } a(). The unt ndces Q 0 and I() are defned n 2 and bounded by (5.1) and (4.1). Further, w = 1 unless k k( 1 ) 0K when w = w 2 ( 0 )/w 2 ( ) for the unque subgroup S 0 whose fxed feld contans k( 1 ). Let C() be the subgroup of the deal class group of K composed of classes wth orders prme to [ : 0 ]. Then the part of the class number relaton (6.2) prme to [ : 0 ] s nduced by the drect sum decomposton C( 0 )/C() = C( ) / C( ) S 0 gven by γ = [: 0 ] /0 γ for γ C( 0 )/C() and the natural dentfcaton of C() as a subgroup of C( 0 ). Proof. Defne w by w a0 = w 2 () a(). Then Theorem 2.3 of [9] gves w = 1 f k or 1 0 K. Otherwse, f J s the alos group of K/k( 1 ) then (1.6) yelds ( J / J / ) = J / J /. Consequently J / J / for at least one S 0, say, and J for such a subgroup. owever, f J also contans S 0 then and so =. As w 2 () = 2 for all S 0 except = the value of w s w 2 ( 0 )/w 2 ( ).

9 Kurodas class number relaton 49 Now a()r() = 0 s apparent from equatng the ranks of 0 U and U 0. ence (6.2) s obtaned from (2.2), (2.5) and (3.7). The class group relaton holds because the norm equaton gves γ = [ : 0 ] /0 γ. A partcularly useful specal case of ths theorem s a generalzaton of Kurodas result [5], whch ncludes the formulae of erglotz [4] and Parry [6]. (6.3) TEOREM. Let p be a ratonal prme, n 2 an nteger, and a (1 n) elements of a number feld k. Suppose k * = k ( p a 1 n ) has degree p n over k and let {k t t T} be the set of (p n )/(p) subfelds of degree p over k. Denote by h, h t, h k ; U, U t, U k ; and W, W t, W k the class numbers, unt groups, and groups of roots of unty of k, k t, and k respectvely. Set Q = [ U : W t T U t ]. Let u be the number of algebracally ndependent felds k t of the form k(ε) where ε p U k. If one of the k t s k( 1 ) let v satsfy 2 v = w 2 /w 2 where w 2 and w 2 are the 2-components of the numbers of roots of unty n k and k( 1 ). Otherwse put v = 0. Let r, r t, and r k be the -ranks of U /W, U t /W t, and U k /W k. Then where A = h* hk h = Qp A k ht t T n u p p ( n )( r 1 * 1) ( rk + u v. 2 p p 1 2 ) The ndex Q dvdes p B for B = B t = (n)(r r t +1) and any feld k t ; and the p n -th power of every unt of k les n W U t. For Ω = k, k t, or k let C(Ω ) be the natural embeddng nto C(k ) of the part of the deal class group of Ω formed from classes whose orders are prme to p. Then there s a drect sum decomposton C(k )/C(k) = t T C ( k ) / C( k). Remarks. The same theorem holds more generally provded only that the alos group concerned s somorphc to the one here. t

10 50 C. D. Walter When appled to dfferent relatve extensons wthn k /k the theorem produces all relatons between the class numbers of ntermedate felds whch can be deduced from relatons between nduced prncpal characters. The value of B cannot n general be mproved beyond B = 1(r +1)(n ) 1(r k +1) ((p n )/(p ) ) because p B s the value of Q when U /W k s somorphc to the lattce L of (2.1). Proof. Example (1.7) gves a relaton between the alos groups of the felds k ( p 1 ), k t ( p 1 ), and k( p 1 ), and Lemma (1.8) allows ths to be lfted to a drect norm relaton between the groups of the felds k, k t, and k. (6.2) gves the requred relaton once the followng equaltes are proved: (6.4) w = p v, (6.5) [: 0 ] a()(r())/2 = p a * x for x = 1(n )(r ) 1{(p n )/(p ) }{r k }, (6.6) Q 0 a * I() a() = Q a * p a * y for y = (p u )/(p ) u. The frst s trval and for the second note that [: 0 ] a()(r())/2 = ([: 0 ]p 1 n ) a()(r())/2 = p a * x. By (2.6) the last s equvalent to a I * 0 I() a() = I( 0 ) a * p a * y, that s, I 0 I t = p y t T n the obvous notaton. By (4.2) the ndex I t s 1 or p. If k(ε 1 ) and k(ε 2 ) are two of the k t wth ε p 1 and ε p 2 n U k then k(ε 1 ε 2 ) s k or another such k t. Thus f there are u algebracally ndependent such felds then the total number s the number of subfelds k t of ther composton k c, vz. (p u )/(p ), and, by (4.2), I t = p y+u δ where δ = 0 or 1. Precsely, δ = 0 f no feld k t has the form k(ω) where ω W has p-power order, or f one of the k t s k( 1 ) and t has correspondng ndex I t = p. Otherwse δ = 1. Let us suppose that f one of the k t s k( 1 ) then I t = 1 for t. The lnear combnatons of u algebracally ndependent generators ε of the k t wth ε p U k generate each U t «U k over W t U k and so generate

11 Kurodas class number relaton 51 U 0 «U k over W 0 U k. No lnear combnaton whch s not a pth power can le n U k because of ther algebrac ndependence. Therefore any combnaton n W 0 U k les n the equvalence class modulo U k of a root of unty ω k wth ω p k and yelds a subfeld k t of k c wth I t = 1. Conversely, such a k t n k c leads to a lnear combnaton n W 0 U k. ence I 0 = p u δ. Now suppose that k = k( 1 ) s one of the k t and that I = p. Then the u algebracally ndependent ε generate each U t «U k over W t U k except when t =. Thus they generate over W 0 U k a subgroup of ndex p n U 0 «U k. On the other hand there s a lnear combnaton of them whch les n the same class modulo U k as 1. Thus agan I 0 = p u δ and I 0 I t = p y whch proves (6.6). The bounds on the order and exponent of U /W U t come from (5.3) and from applyng (1.6). The frst remark s clear; for the second see [8]; and for the last use (3.7). References [1] E. Artn, Lnear mappngs and the exstence of a normal bass, Volume for Courants 60th Brthday, Interscence, New York [2] R. Brauer, Bezehungen zwschen Klassenzahlen von Telkörpern enes galosschen Körpers, Math. Nachr. 4 (1951), pp [3]. asse, Über de Klassenzahl abelscher Ζahlkorper, Akad. Verlag, Berln [4]. erglotz, Über enen Drchletschen Satz, Math. Zetschr. 12 (1922), pp [5] S. Kuroda, Über de Klassenzahlen algebrascher Zahlkörper, Nagoya Math. J. 1 (1950), pp. 10. [6] C. J. Parry, Class number formulae for bcubc felds, Illnos J. Math. 21 (1977), pp [7]. P. Rehm, Über de gruppentheoretsche Struktur der Relatonen zwschen Relatvnormabbldungen n endlchen alosschen Körpererweterungen, J. Number Theory 7 (1975), pp [8] C. D. Walter, Class number relatons n algebrac number felds, Doctoral thess, Cambrdge Unversty, [9] Brauers class number relaton, Acta Arth., ths volume, pp DEPARTMENT OF MATEMATICS UNIVERSITY COLLEE Be1feld, Dubln 4, Ireland Receved on and n revsed form on (814b)