On the Invariants of some Zℓ-Extensions Citation Bloom, John Roll (1977) On the Invariants of some Zℓ-Extensions. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/s74r-gx65. https://resolver.caltech.edu/CaltechTHESIS:11102025-193024391 Abstract Let k be a number field, l a prime, kck 1 ck 2 c...cK, and kcm 1 cm 2 c...cM two Z l -extensions of k. The structure of the galois group of a certain extension of MK is studied, and it is shown how, in some cases, the l-parts of the class groups of the intermediate fields m i k j . can be obtained from this group. This galois group is a module over Z l [[S,T]], the power series ring in two variables over the l-adic integers, but the structure theory of such modules is not well developed. The main results come from studying the structure of this group as a Z l [[S]] or Z l [[T]] module. Necessary and sufficient conditions are given for this group to be a Noetherian module over Z l [[T]], and thus it has a well known structure. Sufficient conditions are given for the module to be a torsion module. The structure of this group is then used to obtain information on the Iwasawa invariants μ and λ of the Z l -extensions km i /m i and Mk j /k j . In suitable situations it is shown that μ(K/k)=O implies that μ(Km i /m i )=0 for all i, and λ(Km i /m i )=rl i + i Σ j=0 c j φ(l j ), with c j =0 for all j>n 0 and it is shown that r=0 iff the above module is torsion. In certain situations, this group is also used to study the invariants of all Z l -extensions of k contained in MK. With suitable hypotheses, it is shown that at most one Z l -extension has μ≠0. Some examples are computed. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Kisilevsky, Hershy Harry Thesis Committee: Unknown, Unknown Defense Date: 1 January 1977 Funders: Funding Agency Grant Number California Institute of Technology UNSPECIFIED Ford Foundation UNSPECIFIED Record Number: CaltechTHESIS:11102025-193024391 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:11102025-193024391 DOI: 10.7907/s74r-gx65 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17753 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 13 Nov 2025 22:04 Last Modified: 13 Nov 2025 22:25 Thesis Files PDF - Final Version See Usage Policy. 11MB Repository Staff Only: item control page

ON THE INVARIANTS OF SOME ~£-EXTENSIONS Thesis By John Roll Bloom In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy California Institute of Technology Pasadena, California 1977 ii ACKNOWLEDGEMENTS I would like to thank my wife Janet, and my parents for the love and support they have given me. I became interested in this topic through my advisor, Dr. Kisilevsky. I appreciate the help he gave me, and the opportunity I had to work with him. I received financial support from a Ford Foundation Fellowship, and through various fellowships and assistantships granted by the California Institute of Technology. iii ABSTRACT Let k be a number field, £ a prime, kck ck c ... cK, 1 2 and kc m c m c ... c M two Z£ -extensions of k. The structure 1 2 of the galois group of a certain extension of MK is studied, and it is shown how, in some cases, the £ -parts of the class groups of the intermediate fields m. k. can be obtained from l J this group. This galois group is a module over z1 [[S,TJ], the power series ring in two variables over the £-adic integers, but the structure theory of such modules is not well developed. The main results come from studying the structure of this group as a ii [[SJ] or Zi[[T)] module. Necessary and sufficient conditions are given for this group to be a Noetherian module over 'll£ [ [ Tl j, and thus it has a well known structure. Sufficient conditions are given for the module to be a torsion module. The structure of this group is then used to obtain information on the Iwasawa invariants µ and A of the 2 0 -extensions ;:, Km./m. and Mk./k .. l J l J In suitable situations it is shown that {L(K/k)=O implies that µ(Km . /m. )=O for all l i, and A(Krn . /rn.)=r!i + l l l .t c . m(£j), with c.=O for all j >n 0 J J= 0 JY and it is shown that r=O iff the above module is torsion. iv In certain situations, this group is also used to study the invariants of all ~£-extensions of k contained in MK. With suitable hypotheses, it is shown that at most one ~i -extension has µ*O. Some examples are computed. V TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ii ABSTRACT iii Chapter I. INTRODUCTION . . . . 1 II. FURTHER DEFINITIONS AND NOTATION . . . 5 III. COMPOSITION WITH A FINITE EXTENSION . 8 IV. COMPOSITION WITH A '.l,e -EXTENSION . . 13 V. OTHER 'l1,e -EXTENSIONS CONTAINED IN THE COMPOSITE . . ..... 26 SOME EXAMPLES . 36 REFERENCES . . . . . . . 43 VI. 1 I. INTRODUCTION Let k be an algebraic number field and£ a fixed rational prime. We shall be concerned with a certain type of extension of k, called a r-extension, or a ZR-extension. Let~£ denote the ring of £-adic integers, and ~; the £-adic integers considered as an additive group. Let K be a field containing k pot necessarily a finite extension of the rationals, Q), then we say that K/k is a ~-extension if K/k is normal and g(K/k) ~ ~;, where g(K/k) denotes the galois group of the extension K/k. Let r=g(K/k). rn=£n I', n=0, 1,... . r has as subgroups of finite index Let k n denote the fixed field of r . n Then k=k ck c ... cknc ... cK.co The k are the only subfields n 0 1 of K containing k, and K=LJ k . It can be shown that n=l n k /k is cyclic of order £ n. The field k is called the n n n-th layer of the 2e-extension K/k. An example of a ~£-extension is constructed in the following fashion. Let ( denote a primitive (£n+l)-st co root of unity. Let kn=Q((n), K=1) kn. Then g(kn/k ) ~ ~/e n z , 0 1 n and g(K/k ) ~ 1~ Z/£ ~~ 2£. If £=2 we take k instead of 0 1 n k as the base field. For£ odd, K/k is a ~£-extension, 0 0 and furthermore K is normal over Q. Then g(K/Q) contains a unique element of order £-1, and g(K/Qh~i!).6., with l.6./=£-1. Let P be the subfield of K fixed by .6.. P/Q is a ~e -extension of Q, in fact the only ~£-extensio n of Q. For £=2, we take 2 P to be the subfield of K fixed by complex conjugation. For any number field k, the extension kP/k is a i,e-extension, called the basic, or cyclotomic, 7/i,e-extension of k. An algebraic number field always has at least one Z,e-extension, the cyclotomic one. For a number field k, let M denote the composite of all Z,e-extensions of k. g (M/k)"'Z'1 for some integer a. If k has 2r 2 Then distinct complex imbeddings into the complex numbers then l+r 2~a~[k:Q] [7, p.253]. The conjecture that a=l+r 2 is equivalent to Leopoldt's con- jecture on the nonvanishing of the .t-adic regulator. For any algebraic extension of Q, F, let F denote the maximal abelian unramified .£-extension of F (i.e., F/F is unramified and g(F/F) is a profinite t-group. In general, a normal extension L/F is unramified, if for each valuation v of L, the inertial group, T ={aEg(L/F) I v(a(a)-a)>O'v'aEL} V is trivial). Let AF=g(F/F). For Fa number field, Fis a subfield of the Hilbert class field of F and AF~ the Sylow £-subgroup of CF, the class group of F. For the ?.£-extension K/k, kck c ... cknc ... cK, the 1 groups Akn are related in an interesting way. Define e n by ten= iAkn I. THEOREM (Iwasawa). v such that en=µ.tn+:\n+V 'v'n?n µ, "tt:i:o. There exist 0 integersµ, A, and for some integer n 0 =n 0 (K/k), with 3 The integers µ = µ (K/k) and A=A (K/k) are called the (Iwasawa) invariants for the ~p- extension K/k. A natural question would be, £or a Zp-extension K/k, how does bne determine the invariants u, A and the integers v and n . 0 The main interest has been in the invariantsµ and A, and although methods for the determiningµ and A are not known in general, there are results for certain special cases. For the cyclotomic ~-extension of the field of R-th roots of unity, the invariants can be computed via the R-adic L-series (modulo Vandiver's conjecture), with the connection based upon a theorem of Stickelberger. Greenberg [3] has some results on the basic extensions of totally real fields. A number field which is not totally real has infinitely many ~p-extensions, and the question is still open for the non-cyclotomic extensions. In fact, the set of ~£-extensions of a number field has not been canonically indexed, so it is difficult to work with a specific non-cyclotomic ~£-extension. Carroll and Kisilevsky [2] have, in certain situations, characterized a set of ~£-extensions of k, which are independent, and together with the cyclotornic ~£-extension, generate the composite of all ~Q-extensions. These independent ~ 2 -exten- sions are the unique ~Q-extensions normal over Q. For this case, certain congruence relations in the invariants were found, and a functional equation was given for a certain 4 characteristic polynomial. (The polynomial uniquely de- termines the invariantsµ and A.) This thesis investigates the situation where a :z- 2 -extension is described in terms of one or more "known" ~ 2 -extensions. The first situation investigated is the composition of a ~ -extension K/k with a field of F to 2 obtain a z2 -extension KF/F. (c.f., Iwasawa [6]). The same situation is considered where F runs through the various finite layers of a '.l 2 -extension M/k. 5 II. FURTHER DEFINITIONS AND NOTATIONS Let K/k . be a ZR-extension. Let K be the maximal abelian unramified £-extension of K. Let G = g(K/k) I X =AK= g(K/K). Then Xis a normal subgroup of G, and G/X~f=g(K/k)~2£- r acts on Thus X by conjugation, and ~also acts naturally on X, with the action of a E ZR given by x,-<x::a = lim xan _ (This an-..a limit exists since Xis a profinite Q-group.) Both these actions are continuous. These actions allow us to consider X as a module over the group ring ZQ[r]. X can also be considered as a module over the ring of (formal) power series in one variable, ZR [[TJ]. This action is obtained by picking a topological ' generator Y 0 of r (i.e., < y 0 > is dense in r), and defining the action of Ton X by Yo x = (l+T)x for all ZQ [(l+T)] is of z dense in 2[[TJ] on x. denote XEX. Since z2 [[TJ], this defines a unique action c.f., [ 7 ]. From now on, we will let AT ZQ [[TJ]. In general any compact profinite £-group on which r acts continuously admits a continuous action by AT. c.f .[7] A few elements of AT are defined for future use . .n DEFINITION. wn(T) = (l+T)£ -- 1, n~O. v n,m (T) = wm(T)/wn(T), mzn>O. 6 DEFINITION. Given two AT modules X and Y, a homo- morphism f:X -Y is a pseudo-isomorphism if the kernel and co-kernel are finite. We say Xis pseudo-isomorphic to Y if there exists a pseudo-isomorphism ;f;: X ---- Y and write Pseudo-isomorphism is not, in general, an equivalence relation. That is, we can have X = Y but Y ~ X. p p isomorphism is, however, transitive. Pseudo- Also, if X and Y = = are Noetherian torsion /\T modules, then X p Y implies Y p X. . ... , e s be non-negative integers and p 1 , ... , ps be prime ideals of height one in AT. Each p. l is either (Q), the ideal generated by Q, or (f. (T)), the l ideal generated by a distinguished irreducible polynomial (A polynomial f(T) is distinguished if, f(T) = Td + fg(T); d>0, d>deg g(T) .) Define the module, Such a module is called an elementary AT module. Every Noetherian AT module is pseudo-isomorphic to a unique Also, Xis a elementary module Ex· Noetherian torsion AT module if and only if the associated elementary module E(e 0 For a ZR -extension of a number field, K/k, the module X = AK is always a Noetherian torsion AT module. 7 If we write, X~E p then the invariants µ(K/k), i\(K/k) are as follows: m i\ = i~l qi deg f i. The above description ofµ and i\ also allows us to characterize them in the following way. vis a vector space over QR, and dimQ o . 22 2 Let D denote Let V= x® V = i\. Q the kernel of the map, R X. X - X f-+- Qx Thenµ= 0 if and only if Dis finite. At this point we also wish to mention the following fact, a consequence of ramification theory. If Fis an . abelian extension of a number field k with g (F/k) ;;:;; 2; = ZR @ R. ... 0 ZR, then F/k is unramified outside primes above The case F/k a z2 -extension is of particular interest. 8 III. COMPOSITION WITH A FINITE EXTENSION Let k be a number field, K/k a ~_e -extension, and r F /k a cyclic extension of degree ,£ , k = F Fr=F, with c ... c 1 Assume furthermore that FnK = k. [Fi:k] = .ti. O c F Since g(KF./k) = g(KF ./F.) 0 g(KF./K)"- Z 0Z/.eiz, l l l KF./F. is a ~o-extension for i = 0, l l 0 l "" -"' ... , r. We will obtain certain relation~ among the A-invariants of the various extensions KF./F .. l see [ 6 For the relationship of the µ-variants, l J. Let X. = g(KF./KF.). l l Then A(KF./F . )=dimQ (X. ®'7 Q). l l a Let g(KF/K) =<O" >. .tr ,£ l l ~ £ = 1. LEM.MA 3.1. Q) = dimQ ,£ -,e If each prime of k which ramifies in F/k is finitely decomposed in K/k, then X . .:: p 1 i X /(a,e, -l)X. r r To simplify the formulas, we will let D(X)= dimQ,£ (X ®:?:t Q.e) for any AT module X. Th: first statement of the lerruna now reads D(X.) = D(X /(a,e, -l)X). 1 PROOF OF LEMMA. commutator I G .. l r We construct the following fields. I Let E. = KF KF .. l r Let G. = g(KF/KF.), and let G. be the l l l subgr9up of Gi. l Let M. be the fixed field of l The field M. is the maximal subextension of KF/KF . l l which is abelian over KF .. l The following relationship holds 9 among these fields: This provides us with an exact sequence. O-g(M./E.)-g(M./KF)-g(E./KF)- 0. l l l l If we form the tensor product of each element of thesequence with Q£, and take the dimension of the resulting vector space, we get, D(g(M./KF)) = D(g(M./E . )) + D(g(E./KF)). l l l l Since D(X) = D(g(KF/KF))is finite, each term in the above r equation is finite. , The first part of the lemma will be done if we show D(X.) = D(g(E . /KF)), D(g(M./KF)) = l i l l D(X /(a£ -l)X ), and D(g(M./E.)) = 0. r r 1 First, g (E. /KF) l 0 - 2: g ( KF . /KF n KF . ) l 1 g (KF. /KF n KF . ) . l g(KF./KF.) - l We also have l l g(KFnKF./KF . )-0. l l l Since the last term in this exact sequence is finite, we can actually conclude that g (E. /KF) = X. , so the relationp l l ship D(X.) = D(g(E./KF)) is proven. l l To prove the second relationship we argue as follows. I By the definition of M., g(M./KF.) = G . /G. l l l , l I therefore q- (M 1. /KF) = Xr /(G.1 n Xr )· __ We can take a erator of g(KF/KF.). l Since X r Ql l , and as a gen- Let a= a 2 i denote a lifting of and a generate G., and X 1 r is an 10 • a b e l ian group, one sees t h at G'. = < a x a-l x-l I XEX > = l r i (aQ -l)X . r i , Thu~ we have g(M./KF) 2= X /(aQ -l)X, and therefore 1 r i r D(g( M. /KF)) = D(X /(aQ -l)X ). r 1 r = 0. Finally, we show that D(g(M./E.)) l l Since g(M./E.) c g(M./KF.), it suffices to show that D(g(M./KF.)) l l- l l l l = O. } be the set •of valuations of KF l. which ramify Let { w. J in M./KF .. l W. J Let {T.} be the set of inertial group for the l J in M./KF . . l Since KF.c KF. s_M., and KF. is the maximal l 1- l l 1 unramified subextension of M./KF., g(M./KF.) is precisely l l l l the group generated by the T .. T.I KF is the inertial group for w. in KF/KF .. Qr-ii Then t. J J SO Let t .ET.. J l Qr-i t. E T.ng{M./KF). J J l J l - 1, and therefore, J __ Qr-i _ l is the identity KF Now M./KF is unramified, so J Thus t. l g {M. /KF.) - 1, and g(M./KF.) l . Qr-i _ T . n g (M . /KF) = 1 . J J l l @rn /LJQ Qn x, = 0. This completes the proof of the first statement of the lemma., The second statement now follows quickly. We have already shown that, g(E./KF) 2= X.' and g(M./KF) 2= X /(a.ei _l)X p l l r l r We also have the exact sequence, 0-g(M./E.) l l g(M./KF) l g(E./KF) l 0. We will be done if we show g(M./E.) is finite. l show that g(M./KF.) is finite. l l l We actually 11 It was shown above that g(M . /KF.) is the group l l generated by the inertia groups { T . }, and each T . is finite. J J Let v. be the restriction of W. to F .. J J in KF/KF., l l vJ. must ramify in F/F .. Since w. ramifies J Since F/F. is a finite l l extension of an algebraic number field, there are only finitely many ramified primes. The hypothesis that each prime ramifying in F/k is finitely decomposed in K/k implies v. to KF .. that there are only finitely many extensions of J l Thus {T.t is finite, and g(M./KF.) is finite. □ Jf • l l This lemma can now be used to relate the A-invariants for the various Zn-extensions KF./F .. x. l l hence PROOF. The characterization of A(KF./F.) in l section II is that A(KF./F.) l that D (X.) l l = D(X.). 1· l The above lemma shows = D(X r /(a£i_l)X). r The map Let V be the On vector space X ® QR. x. r Z£ is a linear map on V. Let V. denote the null J Now, dim dim -o V R 0£ (X r ®z Q) '£ R - dim 0R V i" 12 Therefore, we have, V (o-Q i-1 i dimQ Q V. l = l- -l)V. l = (o-.tni-1 -1) V l.. Let W. l ~(Qi) dim Let ~(t) 0R W.. l i If W. = O, we are done. Assume W.* 0. l = t~ -1/tg nomial in QQ[t]. We are done if we show i-1 -1. l ~(t) is an irreducible poly- Since W.*O, but ~(o-)W. l l minimal polynomial for a on Wl.. , = 0, ~(t) is the Therefore, P(t), the characteristic polynomial for a on w., is a power of ~(t). l P(t) = (cp(t))s. s. ~<ei). □ Thus dimQ Q W. l = deg P(t) = s deg ~(t) = 13 IV. COMPOSITION WITH A ~.,e-EXTENSION Let k=k ck c ... ck, and k=m c m c ... cM be two disjoint 0 1 0 1 ~.,e-ex t e nsions of k. The invariants of the Z.,e-extension Mk./k. can be related to the invariants of the ~.-extension l l ,,:, M/k b y Theorem 1, but in this case, stronger results hold. We therefore study these extensions. Since not every number field has two disjoint Z,e-extensions, it is necessary to assume that k has at least one complex imbedding. For the remainder of this thesis, let k be a number field with at least one complex imbedding. Let k k. be the cornposi te of all Z..e -extensions of 2"..e Then G=g(k~..e/k)~ ~;, with a ? 2. It is possible to pick HcG with G/H~z,e0 Z.,e, and if N is the fixed field of H, One can pick r 1 and r 2 contained in g(N/k) such that r "" r "" ~.£' and g(N/k) = r1 0 f2" 1 2 Let K be the fixed field of field of r2 . r1 , and M the fixed Then K/k and M/k are disjoint ~-extensions of k, and N=MK. Let N be the maximal abelian unrarnified .£-extension of N, and XN=g(N/N). We will show that XN is a module over ~ [[s,TJ] in a natural way. Let a be a topological generator of T be a topological generator of r 2 . r1 , and let Since g (M/k) ~ r 1 , with the isomorphism given by restriction,g(M/k) is topologically generated by the restriction of a to M, which we 14 also call a . Similarly, g (K/k) is generated by T . n the n Let Nn .. denote the subfield of N fixed by <O'.t' , 7.t' >, n closed subgroup generated by a.t' and T ,en_ Th en No= k / 00 00 • and N= LJ Nn . n=l - _LJ - [ Since N -n=lNn (c.f. 10 ]) , g(N/N)=lim g(N/N). - n Each g (N /N) is a finite £-group, and therefore XN is a n n profinite £-group. r 0 r acts ' Let G = g(N/k). 1 2 on XN by conjugation, and thus one obtains a continuous r2 J on XN. action of Zj, [ r1 0 It is not hard to show that the correspondence l+T----T, l+s--a, provides a continuous (c.f.[ 4 THEOREM (Greenberg). ]). XN is a Noetherian torsion ~,e[[s,T]J module. The structure of z,e[[s,TJ] modules is not as well classified as the structure of z,e[[sJ] modules. The ,e-part of the class group of the intermediate fields k.m. l J (ki cK, mj c M) is connected with the z,e[[s,T]J structure of XN. The following proposition describes the connection in the particular case where N/k has a unique totally ramified prime. PROPOSITION. If N/k has a unique totally ramified prime, then, g ( k . m . /k . m . ) l J l J XN/<w. (T)XN, w . (S)X >. l J N n 15 PROOF. Let i and j be fixed. Let E denote the largest subfield of N which is abelian over k . m .. l NcEcN. l J l J wJ. { S ) XN >· L Let G = g(N/k.m.). l I Then We will show that g(k.m./k.m.) . .__ g (E/N), and that g (E/N) "' x_ j < w . ( T) XN , •N l G . J Then Eis the fixed field of J We will show that G I = <wi (T)XN, w . ( s) xl\J >. J L Let fi denote the prime of k.m. which is totally J l ramified in N/k.m., and let T l J V be the inertial group in N/k.m., for a valuation v of N which extends the valuation l J induced by fi . Since N/N is unramified, TvnXN=0. is the inertial group for fi in N/M, T XN=G. V r Since TvXN/XN We have G/XN"' Tv, and Tv acts on XN by conjugation, so G is a semidirect product of XN by TV . Thus G , = <txt -1 x -1 j tETV, XEXN .> = < XNt-1 I tt:T V >. We can also identify TV with g ( N/k . m . ) . l then, t-1 so, J If , 16 so I G Thus, Nf._Es;_N, with G Therefore, g(E/N) ~ XN/G 1 1 = g(N/E) s;_ g(N/N) = XN. = XN/<Wi (T)XN, wj (S)XN>. Let ft be the prime of k.m. which ramifies in ]. J N/k . m., and let T4 be the inertia group for ft in E/k . m .. J. J r l J Since Tf- projects onto g (N/ki mj) and is disjoint from g (E/N), we have g (E/k .m.) ~ g (E/N) 0 l J -· T4 . r Since k . m. cE and the fixed field of J - 1 tained in k.m., we have E = N k . m .. J l • l J J l J r is con- Therefore, g ( E/N) "' g ( k . m . /k. m . ) , and we are done. l T _✓, D We now return to the study of XN. Since the structure of ~.e[[s,TJ] modules is not well known, we notice that the action of 'ltf,e [[s,TJJ on XN provides an action of z_e[[sJ] and Z,e [[T]] on XN, and study the structure of XN as a module over these rings. THEOREM 2. If t is finitely decomposed in M, then XN is a Noetherian z_e[[TJ] module if and only if µ(M/k) = 0. Similarly, if t is finitely decomposed in K, then XN is a Noetherian z.e[[sJ] module if and only if µ(K/k) = 0. PROOF. We will prove the first statement. second follows from a change in notation. The 17 The module XN is a Noetherian 'l,,e[[TJ] module if and only if XN/(T,£)XN is finite. The module XN/(T,.t)X N is finite if and only if XN/TXN has a finite rank. Let E c N be the fixed field of TXN. H = g(E/N) XN/TXN. Then Since TXN is the commutator subgroup of g(N/M), G = g(E/M) is abelian. Therefore we can form the factor group G/H, and G/H ~ g (N/M) ,.._ 'li,e. Thus XN is Noetherian if and only if G has finite rank. Let I denote the subgroup of G generated by the inertia groups of all valuations of M. Since only valu- ations above ,e can have non-trivial inertia groups, and we have assumed there are only finitely many valuations above £, I has finite· rank. Hence G has finite rank if and only if G/I has finite rank. Let L be the fixed field of I. Then Lis the maximal abelian unramified extension of M contained in N. Since M ~ N, L=M. Therefore G/I ~ g(M/M). Finally, XN is Noetherian if and only if g(M/M) has a finite rank, which occurs if and only if µ(M/k) I = 0. CJ = k 0 ck 1 c ... ck, and k = m0cm 1 c .. ~cM are two disjoint ~,e-extensions of k, and N = MK. For each i~O, let F. = k.M. Then F./k. is a ~,,-extension and we i l l l V Recall that k now study these extensions. Let X. l group of = g(F l. /f.). l The structure of the class k.m, then-th layer of F./k., is determined l n l l by the structure of X. as a~ l S module. That structure 18 is not easily determined in the general case. We will find information about the invariants of F./k . from the l l AT structure of · xi. Recall that XN = g(N/N). THEOREM 3. PROOF. If N/M is unramified, then, If N/M is unramified, then NcF.cN, and 1- F. is the maximal abelian extension of F. contained in N. l l Let G. = g(N/F.). l Then X. = g(~/F.) l l l l ~ Gl. /G~. l Since G. l i is a semidirect product of XN by g(N/Fi) = <T,e >~~.£' we Therefore X. is a semi1 direct product of XN/ wi (T)XN with g(N/Fi) ~ ~,e- But since X. is abelian, the product must actually be a direct l product. 0 COROLLARY. If µ(M/k) = 0 and N/M is unramified, i then µ(F./k.) l l = 0 and A(F l. /k.) = r£ l numbers rand c n i + n~Ocn~(,en). The are non-negative integers, and c =0 Vn~n , for some integer n . 0 0 n The integer r is zero if and only if XN is a torsion AT module. PROOF. If µ(M/k) AT module, by Theorem 2. = 0, then XN is a Noetherian Therefore XN is pseudo-isomorphic to some elementary module E, and XN/wi(T)XN; E/wi(T)E. Thus we have • X.l ~ E/w.1. (T)E 0 Zb. p ,c, 19 If, m h E = A ~ j (£i AT/ ( f j ( T) ) n j gcp1 then, h gcpl AT/(wi (T) ,.tng) 0 The module A /(w. (T) ,.enh) T i, nh. l 'lf.e- is a finite for each . The module AT/(wi (T), (fj (T)nJ)) is either finite, I f. (T) { w. (T) , or is pseudo-isomorphic to AT/v J l for some n_s;_i, if f. (T) J Let c Then C n n = V n- l , n.(T). n- 1 ,n (T) Define V l O (T) - , be the number off . (T) equal to V i J if . = T. n- 1 , n (T). = 0 Vn:::::n 0 , and X.::: (AT/(w . (T))r 0 · (AT/v (T))cn0~ 1 p 1 n= 0 n- 1 ,n L 0 Since the module on the right has no elements annihilated by .t, X. has only finitely many elements annilated by .e, hence l µ(F . /k.) l l = 0. We also have, A ( F. /k. ) l l = dimQ,e ( X.l ®z'£ Q ) This can be made into appropriate form by change of c . 0 Remark: If A(M/k) THEOREM 4. = 1, then r = 0, D thus A(F./k.) is bounded. l l If N/M is ramified at some valuation, and unramified at all but a finite number of valuations, then there are integers n 0 and n , and AT submodules Y.1c- XN 1 • 20 such that "iji~n 0 , Xi~ XN/Y i, and 'iii< n 0 , xi ; XN/Y i. = vn ,1. (T)Yn1- Furthermore, 'v'i~n , Y. 1 1 1 Remark: A more exact description of n , n , and the Yi is 0 1 given in the proof. Remark: Both the statement and proof of this theorem are essentially the same as in the case of a Z~-extension of a number field, and closely follow Serre PROOF. 10 ]. Recall that F. is the maximal abelian l unramified !-extension of F .. The extension NF./N is l l unramified, so NF 1. -c N, and therefore F 1. -c N. G. l = g(N/F.), and H . = g(N/F . ). l l l X. Then, g ( ~/F . ) ~ G . /H .. - l Let l l l l The theorem will follow from a description of the H .. l Since G . /H . is abelian, G: c H.. l l l - As l above, Let L . denote the fixed field of G:. l We l The extension L.l /F l. is abelian, and have F 1. -c F.1 -c L 1. -c N. F . is the maximal unramified subextension. l Let VO' - ... , v ramify in N/M, and IO' denote the valuations of N which r • • • I I r Define b. by I . = 1bj groups. J J the corresponding inertia r 2, and let no = min b J. I If necessary, relabel the v. so l Pick valuations w , 0 • • • I V r • ... , w of N which extend r Let I~ denote the inertia group for w. in N/M. J J 21 , Then I j ...... '1£,e. I~, Pick O"o I ••• a I . .. , Ir such that aj IN = T topological generators for r ..eb· J. We have F 1. -c N c- L., with L.l /N unramified, so each l valu2. tion which ramifies in L./F. ramifies in N/F . . l l We l conclude that F. is the subfield of L. fixed by, l l ... , I I r > n g ( L . /F . ) . l 'I'hus we have, r LJ , = < wl. ( T) XN , J= . O I . n G . >. J l H. l = max Let t .. lJ t • J. Let a . . lJ a lJ . . E XN. =O",t . 1 J l o-0 Then I'. n G. ( 0, i-b.) . J J ..,b. -b +t . . 0 J l J l Since a . . IN is the identity, lJ -,c, We have <I: n G . , I' nGi>=<I~n Gi, aij > . J 1, 0 Therefore, I The groups XN and I 0 n Gi are disjoint, and for XN ( I~ n G i ) = Gi • have I~ n Gi acting on XN by conjugation. Therefore G. is a l I semidirect product of XN by I 0 n Gi, and we have, G ./H. l l = G l. / < wl. ( T) X N , I a . l l , ... , a . 1r , IO n G . > ...._ XN/< w . ( T) X l l N , a . l l , ••• , a . 1r > . 22 Let Yi = <wi (T) XN, ail, • • • I a . ir >. We have shown that for i~n 0, xi~ XN/Yi. For i-c:;n , we have F .c F 0 have g(F./F ) == l no 1- G no c F. c L . . no- 1- /<Y. ,Io'>== XN/Y .. l l = X l. . finite, g(F./F ) ~ g(F./F.) l n p l l 0 for i<n , X. ~ XN/Y .. 0 l p l l In this case we Since g(F n0 /F . ) is l Therefore, we have, It remains to show that for i~n , Y.=v . (T)Y . 1 1 n1,1 n1 Since Y . = <w. (T)XN, a . , . . . , a . >, and l l 1 l1 lr w. (T) l = Vn , i. (T) wn 1 (T), it suffic~s to show that, 1 For j = 0, ... , r, let Then by the definition of Y. J = •a n J. y 1 and therefore, y Now, a .. lJ = yf J i-n 1 y-,£ 0 ,.i-n j i-n 1 1 - 0 23 so, Since Y0 IN we have, a .. lJ COROLLARY 1. Theorem 4, µ(F n1 /k n1 If in addition to the hypotheses of ) = 0, then for all i~n +1, 1 i µ(F./k . ) = 0 i i n en cp(,e ) +c, and for non-negative integers r, c, and C n I some integer n . 2 PROOF. Theorem 4 states that X. ~ XN/Y . . l the isomorphism for i~n +l 1 we obtain the e x act sequence, 1) 0 --- Y n /Y i . --- X i• --- X. n --- 0 • 1 1 l From 24 If ,eA = {acA / ta =}O, then µ(Fi/ki) only if ,eXi is finite. = 0 if and Applying the snake lemma to the exact sequence 1), we obtain the exact sequence, is finite, we can show that µ(F./k.) l showing that ,e(Y n /Y.) l 1 l = 0 by is finite. /k ) = 0, µ(M/k) = 0, and by Theorem 2, nl nl XN is a Noetherian AT module. The submodule Y is also a nl Since µ(F Noetherian AT module. Proceeding as in the unramified case, we have, h m y - nl p E = Ar 0 AT/ ( f j ( T) ) n j T j=l 0 g=l AT/(,eng) and therefore, i p(AT/ Vni, i (Tr n~nl +l (AT/'h-1, n (T)) Cn, 2) where c n is the number of f.(T) = v J n- 1 , n (T). Thus ,e(Y n /Y.) 1 is finite. Now A(F./k.) = dim l that for i:2:n +1 1 3) ,\ (F. /k.) l l l . (X. l 0 ,& ®,,, Q.,). u,t, ,£, From 1), we have 1 25 We obtain from 2), en= 0 \in~ n . 2 i Inserting this in 3), we are done. D COROLLARY 2. If N/M has only finitely many ramified primes, and each prime which ramifies is totally ramified, and µ(M/k) = 0, then for all i, i with c n = 0 for all n~n 2 . If, in addition, A(M/k) = 0, then r = 0, so A(F./k.) is bounded. l l PROOF. The first part follows as in the previous corollary, since n 0 = n 1 = 0, and X. ~ XN/w (T)X. n l N 0 If A(M/k) = o, then O = rt +c , so r = c 0 Remark: 0 = o. If r = 0, then XN is a torsion zt[[T]] module, and therefore d = dim 0t (XN ®~ Qt) is finite. It.follows /!Jt that for any 'll,e,-extension L/E, with L~N and L having only finitely many valuations above t, A(L/E) ~ d+l because XL is essentially a factor module of XN0~t· 26 V. OTHER Z£-EXTENSIONS CONTAINED INN We study here an extension N/k, as above, with g(N/k )~ ~£0Zi- Since for any a,b,E Zi , such that£ does not divide both a and b, Zi0Z£/<(a,b)> ~ 2£ , the extension N/k contains infinitely many Z£-extensions. For the next few results we look first at the situation where N/M has a unique ramified prime which is totally ramified, and later at the special case where N/k has a unique totally ramified prime. The examples in Section VI show that the second situation occurs infinitely often, and therefore the first does as well. Recall that N = MK, with k=k c k 0 1 c ... c K, and k=m cm c ... cM two disjoint Z£-extensions, and Fi=kiM. 0 1 PROPOSITION 1. If N/M has a unique totally ramified prime, then Xi=g(Fi/Fi) ~ XN/wi (T)XN. PROOF. This follows from the proof of Theorem 4, and Y.=w . (T)XN. l · l We also use the following lemma [ 9 pg. 155], which is corollary of Nakayama's lemma. We quote Nakayama's lemma for reference. Nakayama's Lemma. module. [ 9 ]. Let R be a ring, and E an R If I is an ideal of R contained in eve ry maximal ideal, and IE=E, then I=O. 27 LEM.MA. ideal . Let (iJ be a local ring and m , its maximal Let Ebe a finitely generated (iJ module, and Fa submodule. If E=F+ ,m E, then E=F. PROOF. We have, E = F+mE = F+m(F+JJiE) By induction E = all n. n F+m E 00 Let D = n,nnE. n=l 2 = F+ni E. for all n, and henc e mD = E - n F~1n .c E J...Or D, hence D = 0 by Nakayama's Lemma, so E = F.O We now combine these two results. THEOREM 5. If N/M has a unique totally ramified prime and g(M/M) is a cyclic~£ module (i.e., then XN = /\.Tr for some rEg (N/N) . Thus XN ~ AT/I, where I is the ideal {h(T}E/\.T I h(T)r = 0 }. PROOF. Let r ' Eg(M/M) generate g(M/M). By Pro- position 1, XN/TXN ~ g(M/M), and the isomorphism is given by restriction. Let r be an _element of XN = g(N/N) such that r IM = r Then XN = 2,e r+TXN .£ f 1Tr+:1n-''<N, where m = (T, .e) 1 • is the maximal ideal of the local ring AT. lemma we see that XN = /\.Tr. Applying the D This characterization of XN allows us to repre s ent the action of /\. 8 on XN in terms of the AT structure of XN. 28 With the hypotheses of Theorem 5, we have XN = ATr. Since SrEXN, there is a power series f(T)EAT such that SR= i(T)r. mod I. The power series f(T) is unique Thus g (S)r = g(f(T))r and g(S)XN = g(f(T))XN. In order for g(f(T)) to be well defined, and for future calculations, we show that f(T) is in (T,.£.), the maximal ideal of AT. If f(T) ¢ (T,,t.), then f(T) f(T)X N = XN" is a unit in AT and Thus SXN = XN, and if XM = g(M/M), we have SXM = XM, since XM is a factor module of XN. But XM is a Noetherian AS module, and Sis contained in the maximal ideal of AS' so we can apply Nakayama's lemma to conclude XM = 0. We have XM = XN/TXNl so XN = TXN, hence XN = 0. If XN = 0, we may assume f (T) e (T, ,£.) . Recall that N = MK, with T generating g(N/M) and a generating g(N/K). There is a 1-1 correspondence between 'Jf.£.-extensions of k contained in N and subgroups H.£.g(N/k) such that g(N/k)/H ~ 'Jf.£.. Such subgroups Hare all of the a b form < T •a > where .£ does not divide both a and b. a b La, b denote the subfield of N fixed by <Ta>{ La, b I a, beZ..e, 1,/a=}.£ -fb} of k contained in N. Let Then is the set of all Z,t.-extensions If .£.fa, then a is a unit in <Taab > -- < (Taab) a-1 > -- <,,..rT'ba-1 >. •v '71.£. , so Th us we can ac t ua 11 y describe the set of ~,t.-extensions of k contained in N as {Ll,b' La,l I a,bEZ,e, ..e laf. Notice that M = Ll,O' and K = L 011 • 29 Assume for the rest of this section that N/k has a unique totally ramified prime, and g(M/M) is a cyclic Z.t module. X = l\ Tr ~ AT/I. N From Theorem 5, there is an rEXN such that Pick f (T) d \T such that Sr = f (T) r. g (T) E .L\T, with g (T) For = 1 (mod ,m) , and aE 'Zl,t, define (g(T))a_ lim (g(T))an_ - a --a n THEOREM 6. PROOF. With the above hypotheses and notation, Since N/L a, b has a unique totally ramified prime, the field NL bis the maximal subextension of N a, abelian over La, b" Since g (N/L a, b) = < Taab >, the subfield a b of N fixed by (Ta -l)XN is precisely NLa,b· Thus we have, -ab a _ b g (La, b /La, b) ~ g (NLa, b/N) ~ XN/ (T a -l)XN ~ XN/ ( ( l+T) (1 +f (T)) -1) XN .L\T/I For La,b* Ll,O' TIL generates a subgroup of a,b finite index in g(La,b/k), so it i s precisely this AT structure in which we are interested. kcL - n-cL a, 1 , with [Ln: k]= .£ n , In fact, for we have, I g(Ln/Ln) ~ AT/(I, (l+T)(l+f(T))b-1, wn(T)). 30 We cannot completely describe L a, the invariants of b/k without knowing I and f(T), but we can determine that µ(L a, b/k) -=I= 0 for at most one of the La, b. Gre e nberg [ 4 ] has shown that µ(La, b/k) is bounded in this situation, by entirely different methods. The result of this thesis assumes a stronger hypothesis, and is more explicit in its result. We first prove the following results. LEMMA. With the above hypotheses and notation, if ,t{((l+T)a(l+f(T))b-1), then µ(L PROOF. b/k) = 0. Assume µ (L a, b/k) =t:- 0. Then if Y=g (L a, b/L a, b), By Theorem 6 we have Y ~ AT/J, where Y/£Y is infinite. J = (I, a, (l+T)a(l+f(T))b-1). Y/.lY We have the isomorphism, (£,J)/(,t) Since AT/(£) has no infinite proper quotients, must have Jc (..e) , and therefore PROPOSITION 2. .t I (1 +T) a ( 1+f (T) ) b -1. D Assume that N/k has a unique totally ramified prime and g(M/M) is a cyclic '.it°,e-module. µ(K/k) -=I= 0, then µ(L a, we b/k) = 0 for all other L a,b. If 31 Since K = LO,l' and µ(K/k) * 0, the lemma PROOF. implies that .el(l+T)O(l+f(T))-1 = f(T). = (l+T)a-1 (mod (.£)). (l+T)a(l+f(T))b~l Z,e, then La, b integer. = Lac, b c • For any a,bE~,e, If c is a unit in Thus we may assume that a is an For a * 0, ,e{(l+T)a-1, and therefore ,e{(l+T)a(l+f(T))b-1. By the lemma, again, we have µ ( L a, b/k) = 0 for La, b * L 0 , l • 0 PROPOSITION 3. Assume that N/k has a unique totally ramified prime and g(M/M) is a cyclic ~,e-module. If µ(K/k) = 0, and La, bnM*k, then µ(L a, b/k) = 0. Since La, b n M ::f:. k, we have £ Ib. PROOF. ~fa, and we may assume a= 1. Therefore It suffices to show that ,e{(l+T) (l+f(T))b_l. We will show that, (l+T) (l+f(T))b-1 T It suffices to show that, We have, (l+f(T)) b = (l+f(T)) .£c = · ( (1 +f ( T) ) c) ,e ,e = (l+g(T)) for some g(T)E(T,£). Now, (l+g (T)) ,e =1 and we are done. 0 = l+g (T) ,e (mod .£ ) .e (mod (.e,T )) 32 Combining these results we have the following theorem. THEOREM 7. If N/k has a unique totally ramified prime, and g(M/M) is a cyclic 2£ module then there is at most one '71£ -extension La, b/k, La, b -c N, such that µ (L a, b/k) * 0. PROOF. done. If µ(L a, b/k) = 0 for all La, b' then we are Assume µ(L b /k) * 0. ao, 0 2 shows that µ(La,b/k) If µ(K/k) * 0, then Proposition = 0 for al~ other La,b· Thus we may assume that µ(K/k) 3, L b is disjoint from M. ao, 0 and since MnK 0 = k = 0. By Proposition we may take K = L b, 0 ao, 0 and N = MK , we may apply Proposition 0 2 and conclude that µ(L a, b/k) = 0 for all other La, b. Remark: (Suggested by Richard Foote.) If N and k were normal over Q, and µ(L/k):f:-0 for some ~£-extension of k contained in N, then any field L I conjugate to ·L would be a '71£-extension of k contained in N, with µ(L' /k):/:-0. If the I hypotheses of Theorem 7 hold, we must have L=L, so Lis normal over Q. This places severe restrictions on L, and in most cases restricts it to a single extension. If µ(L a, b/k)=0 for all La, b' we will show that A(L a, b/k) is bounded. This has been shown in a similar, more general setting by Greenberg [ 4 ]. 33 PROPOSITION 4. If N/k has a unique totally ramified prime, g(k/k) is cyclic, and µ(L a, b/k)=O for all La, b' then \(L a, b/k) is bounded. PROOF . We will assume A unbounded, and construct a ~,e-extension La, b/k with nonzeroµ invariant. Given a field F, k ~F~N, let E(F)={La,blF~ La,b }. F/k is not cyclic, E(F)=¢. finite, If Notice that each field F, with F/k F~N , has only .£+1 extensions of degree .e, F , ... , F,e , with F.1 cN . - O LJ E(F . ). Also E(F) = . 1= 0 l Let F =k. 0 We will pick Fn with [F :k]=..en, in the 1 following manner. Pick F with [F :k]=,en, and\ unbounded in 1 1 E(F ). This can be done since\ is unbounded in E(F ) and E(F) 1 0 is the finite union of E(F . ) for the extensions F . of F of 1, l 1, l degree .e. Pick F containing F n with [F :F • ]=,e, and Aunn n- 1 n- 1 We now show that there is a ~,e-extension bounded in E(F). n -□ La, b' with F n-c La, b for all n, thus La, bn= 1 F n · for at least one We may assume F nM=k. Thus F ~L 1 a1' 1 1 Pick such an a . Pick a so that F cL Since the n n - an,1 1 layer of L is the subfield of N fixed by < Taa,r.en >, we a, 1 see that then-th layer is uniquely determined by the congruence class of a mod .en. Thus we have for i < j, a .=a. l Then a=a n (mod £ n J (mod .£ i ) . Let a=lim a . n . and F n-c La, 1 . ) , One may see this in another way. fixed field of <Ta 1 a,T,en>==<Ta0',T,en >. oo The field F n is the Thus the union of the F C(' is the fixed field of n l < TaO',T..en > = < TaO' >, so lJlF = L . n= n= n a, 1 n 34 In order to . show that µ(L a, )*0, we use the 1 following lemma. LEMMA. If k=F cF c ... cFoo is a Z.e-extension with 0 1 a unique totally ramified prime, µ(Foo/k)=0, A(Foo/k)*0, and n g(k/k ) is cyclic, then rank g(E /E) = min (.£ ,A(Foo/k)). n n PROOF. Let X = g(Foo/Foo). Let Y be a topological generator for g(Foo/k), and make X into a AT module by the action l+T-Y. Since Foo/k has a unique totally ramified prime, g(F/F) ~ X/w (T)X. n n n Since g (k/k) = X/TX is cyclic, X ~l\.T/I. Sinceµ=0, A*0, I=(f(T)), deg f(T)=A. Thus if G=g (F n/F n), G ~ AT/ (f (T), wn (T)) , and G/.tG ~ l\.T/(f(T), wn(T) ,.£) = AT/(T\ T.tn,.e) __ m = min ( A, £ n) . (:l/.£'.l)m with D Returning to the proof of the proposition, each Fn is contained in a ~,e-extension L which satisfies the n hypotheses of the lemma, and A(L/k)>.£ . n Therefore, rank g (F /F ) =.en, so n n I g (Fn /F n )I ->.e.e , If µ(K/k) * 0, then there are certain restrictions on the possible values of A(L a, b/k), given by the following proposition. PROPOSITION 5. If N/k has a unique totally rami- fied prime, g(J\1/M) is a cyclic 'l:.e module, and µ(K/k)*0, 35 then A(La,b,/k) is either 0 or [L a, bnK:k]. PROOF. ,en= [L We can take La,b = L.en,b'' where We have g(r:,--b/L a, a, b) ~ /\..T/(I, (l+T).e a, bnK:k]~ by The orem 6. n (l+f(T) )b-1) Since µ( K/k) * 0, .e !f(T), so, (l+T).tn (l+f(T) )b-1 We also have ( l+T) .en ( l+f (T)) b -1 = T.en = h (T) u (T) (mod e ) for some distinguished polynomial h(T). and unit power series u(T), by the Weierstrass preparation theorem. If d=deg h(T), then Td is the smallest power of T with coefficient not divisible by.£ in the power series, (l+f(T))b-1. Thus d =.en. Therefore g(La,b/La,b) is a factor module of /\..T/h (T), with degree h (T) =.en. If .en= 1, we are done. If A(La,b/k) * 0, then /\..T/(I,~T))is not finite, and /\..T/(I,h(T))= /\..T/p(T) I with p(T)ih(T) and p(T) !I. If n 1 ~ degree p(T) <.e , then we argue as follows. L =k =L n K, If, KcL - n -cL a, b' with [L n :k] = .c-nn, then n n a,b n so rank o(L /L ) = .en. But rank g(L /L )< degree p (T)< .e , - n n n n- a contradiction. Therefore p(T)=h(T), and A(L a, b/k) = .en. 0 36 VI SOME EXAMPLES Let .t be an odd prime, r a primitive .t-th root of unit y , and k=Q(r). k. Let M be the cyclotomic ~£-extension of The extension M/k has a unique totally ramified prime. If .tis a regular prime, i.e., .tfhk, then any ~.£-extension K/k has a unique totally ramified prime, and µ=A=Y=n =0. 0 (c.f.[ 5 ]) . In fact, if N is the composite of two Zi-extensions of such a field k, and X =g(N/N), n then X / (T, S) X ~ g (k/k) =O, whence X =0 by our corollary to n n n Nakayama's lemma. Thus, every number field k.m. contained l J in N has class number prime to£, and every ~.e-extension L has zero invariants. a,b Let .t be such that k=Q(r) has class number exactly divisible by£. Assume furthermore that g(M/M) ~ ~t' where Mis the cyclotomic Z.e-extension of k. for£= 37, 59, 67, and for all primes This occurs .£<30,000 where 8 LEM.MA. (Suggested by David Dummit) If.tis such a prime, M/k the cyclotomic ~£extension, K/k a disjoint ~.e-extension, and N=MK, then N/k has a unique totally ramified prime iff k is not contained in N. PROOF. The field k has a unique prime ft over£. Let v be a valuation of N extending the valuation of k 37 induced by;,. Let T denote the inertia group for v in N/k. Since N/k is abelian, T does not depend upon the choice of v. Let E denote the fixed field of T. The extension E/k is unramified at ft, and since only primes above t can ramify in N/k , E/k is unramified. If klN, then E=k, and ft is totally ramified in N/k. Thus N/k has a unique totally ramified prime. If k~N, then ft is not totally ramified in N/k. fact, since ft is principal ink, In splits in k/k, so N does not even have a unique prime over- ft· D Since k has .£+1/2 independent Z,e-extensions, there are many Zt-extensions K/k such that kiMK. Let K be such a Z,e-extension. Thus we have the situation of Theorems 5 - 7, and can conclude from the results of that section that: 1. Theµ invariant is nonzero for at most one Z,e-extension of k contained in N. 2. (Theorem 7) Ifµ is nonzero for one such Z,e-extension, then .£+1 extensions of k of degree.£ and contained in N must have the following structure. One extension has class group (Z/.£Z).£, and 2 the other.£ have class group Z/£Z or Z/£ Z. One can see this as follows. Since £ is not regular, g (M/M) is not finite [ We have shown that g(M/M) ;? XN/TXN '.:.?' .i\T/(I,T). Thus Ti I. 5 ]. 38 Now g(k/k)~ AT/(I,T,f(T)) 2= ~..e/(f(O)), so f(O)=.tu, with u a unit in "lf,e. We may assume that µ (K/k) :;t:0. g (k /k ) :::=:: AT/ (I, f (T), ( l+T)..e -1). 1 1 Then, Since .e If (T), we have f(T)=£u(T), with u(T) a unit power series, and since µ(K/*0, .elr. Thus g(k /k ) :::=:: .L\.T/(.e,T..e) ~ ('Jf/£'1:).e. 1 1 We also have, g(m /m ) : : : .L\.T/(I,T, (l+f(T)/-1) : : : 2',e/(l+f(O)).e-l : :=: 'lf,e/f 1 ~ 'Z/.e 2 1 2 ~. Each other extension of k of degree.tis contained in some La, l' with g(L a, 1 /L a, ) ~ AT/(I, (l+T)a(l+f(T)-1). Since 1 a f(T)=.tu(T), (l+T) (a+f(T))-1 evaluated at T = 0 is .e times a unit. We have already shown (Proposition 5) that, (l+T)a(1+f(T)) - 1 = (T-c)u (T), where u (T) is a unit power a a series, and ,e Ic. 2 We may now conclude that £ {c. We now have g(L a, b/L a,o , ) ~ .L\.T/(I,T-c), and the .e-part of the class group of the first layer is a factor module of, If the structure mentioned in 2 of the class groups of the degree .e extensions of k did not exist, then one could conclude that µ would be zero for all ~.e -extensions of k contained in N. We now present a few examples for which the invariants can be computed. The theory of Zi-extensions of complex quadratic k is more develop e d than for arbitrary k, and portions of these examples have been computed. 39 by other authors. 1 See, for example, Carroll and Kisilevsky ]. In these examples k will be a complex quadratic number field, k=vCa.. Let N be the composite of all Z', .e-exten- sions of k. Then g (N/ k) ~ '.i¾,.e 0 '11..e . '11,1, -extension of k. Let M/ k be the cyclotomic If ..e is odd, or if .t=2 and all quadratic extensions of k contained in N are normal over Q (it suffices that d have a prime factor congruent to ±3 mod 8.), then there is a unique '71,e-extension K/k which is normal over Q, and MK=N. See [ 1 ] for a proof of this fact. If ..e ramifies or remains prime in k/Q, and the class number of k is prime to ..e, then g(N/N)=O, and µ=>-=v=n =O for every 2',e-extension of k. O Example 1. prime, p= 5 (mod 8). Let .t =2 . and k=Q0, • where p is a Then 2 ihk, 41hk. If Fis a quadratic extension of k contained in N, then F/Q is normal, since p =- 3 (mod 8). Since k is complex, we have g(F/Q) ~ Z/22' 0 ::Z'/2::Z-, so F=kV<j, dE Z' . Because F/k is unramified outside 2, the only possible choices for Fare k(V2), k(vC"2), k(ve--1), and each of these fields is contained in N. Now k(v-J_)/k is unramifi e d and cyclic, and 4{hk. Therefore k cJ-i) has class number prime to 2. prime above z ink is not principal ink, Since ft 2 , the p 2 remains prime 40 in the extension k( ✓-=-i. )/k. Thus N/k( ✓--i) has a unique totally ramified prime over 2, and since the class number of k( ✓-=-i. ) is prime to 2, g(N/N)=O. Thus µ=A=O for every ~ -extension of k. 2 For k ~Fs;_N, we have g(F/F)=O if k(V-l)f.F, and g(F/F) ~ ~/2~ if k (V-1) i F. Example 2. Again 2lhk, 4ihk. Let ,P,=2, p= 5 (mod 8) and k=Q0p. The quadratic extensions of k contained in N are again k (V2), k (V-2), and k (V-1) . Here k = k (V-2) , and the class number of k is prime to 2. Thus N/k(H) has a unique totally ramified prime and g(N/N)=O. have µ=A=O for every ~ -extension of k. 2 We For k ~ F~ N, g(F/F)=O if yC°2EF, and g(F/F) ~ ~/2 Z if \L2¢F. Example 3. Let ,P, =2, p = 3 (mod 8) , and k = V-2p. Then 2lhk, 4ihk, and k(V2), k( ✓--°2), k( ✓ --i) are the quadratic subextensions of k contained in N. We have k = k (V2) and p2 remains prime in k/k. Again, g(N/N)=O, and µ=A=O for every 2 -extension of k. 2 If ks;_F~ N, we have g(F/F)=O if V2EF, and g(F/F) ~ Z/2 2 if V2fF. Example 4. Let i,=2, p = 7 (mod 8), and k = o ✓-"r:>. Assume furthermore that 8 does not divide the class number of o/-2p. These conditions occur for p = 7,23,71, and for infinitely many other p. The class number of k is prime to 2, and 2 splits in k/Q. Let ft 1 and ft 2 denote the primes above 2 ink. 41 Let M be the cyclotomic ~ -extenSion of k. 2 have ks_k (V2)~M, and .f- , ;,, 1 2 ramify in k (\/2) /k, so ;, 1 We and ft 2 ramify totally in M. For kcm - n-c M with [mn :k)=2n, let G = g(mn /k). G I is either 2 n hk or 2 n-1 hkGenus theory implies that I Cm n Let F be the maximal unramified 2-extension of m which n n n n-1 is abelian over k. Then [ F :m ] is either 2 or 2 . Let n n F = 0 Fn . n=l Then F/k is abelian, k _cM_cF, and F/M is not finite. Let G = g(F/k), X = g(F/M) and let T. be the l inertia group for fii in F/k (i = 1,2). Let K. denote the l fixed field of T .. l Since F/M is unramified, T . nX=O for i = 1,2. The l fixed field of XT. is contained in Mand unramified at fi., l and is therefore equal to k. l Thus XT. =G, and G ~ X0 T. ~ l l Therefore, The fixed field of T T is unramified over k, 1 2 thus T T = G. 1 2 We also have X ;: : : G/T i = T /T n T . 2 1 2 Therefore Xis isomorphic to a factor of ~ . 2 But Xis an infinite pro-2 group, so X ~ z , and T n T =0. 2 1 2 Thus F is the composite of all ~ -extensions of k. 2 The fields K and K are dis1 2 join t ~ -extensions of k and F = K K . 1 2 2 Let ki denote the first layer of Ki. Then k , k , 2 1 and k(V2) are the quadratic e x tensions of k . contained in F. 42 We claim that the ramification index of /ti in k k /k is 2. 1 2 Since k/ (V2) k k 2 , and Pi ramifies in k (V2) /k, it is at 1 least 2. Since Pi does not ramify in ki/k, it is at most 2. Thus k 1 k 2 /k(V2) is unramified at 2, therefore un- ramified. One can easily show that k(V2)/Q(V-2p) is unramified. Thus k k /0(V-2p) is unramified. 1 2 The field k k is normal over Q, hence normal 1 2 over Since [k k :o(vC°2p)] =4, k k /0(V-2p) is an 1 2 1 2 abelian unramified extension. We have k k cQ(y2p), whence 1 2 k k = Q(V-2p). 1 2 Using the fact that g(Q(yC°2p)/0(/-=2p)) is cyclic, and that the prime above 2 in Q(V-2p) is not principal, we conclude that primes above 2 do not split in k k /k (V2) . Therefore ft does not split in k /k, and 1 1 2 1 thus does not split in K /k. Similarly, p does not split 2 1 in K /k. 2 Thus K. /k has a unique totally ramified prime. l Since g(k/k)=0, g(K./K.)=0. l l The extension F/K. also has a l unique totally ramified prime, therefore g(F/F)=0. Since F does not have an abelian unramified 2-extension, F = M. We can now describe the invariants of any If L = K or L = K , then µ=A=0. 2 1 Otherwise µ=O, A=l, g (L/L) ~ !% · ~ -extension, L, of k. 2 2 For ks_Fs N, [ F: Q ]<CXJ, we have g (F /F) =0 if F~_ K , 1 n n ] and if Fst,K , g(F/ F) ".':'. 'lf/2 ':£, where 2 = [F:FnK 2 . 1 43 REFERENCES 1. Carroll, J.E., and H. Kisilevsky, "Initial Layers of ~£-extensions of Complex Quadratic Fields, "Compositio Mathematica 32 (1976), 157-168. 2. Carroll, J.E., and H. Kisilevsky, "On Iwasawa's A-Invariant for Certain 'lf£ -extensions, " to appear. 3. Greenberg, R., "On the Iwasawa Invariants of Totally Real Number Fields," American Journal of Mathematics 98 (1976), 263-284. 4. Greenberg, R., "The Iwasawa Invariants of I'-extensions of a Fixed Number Field," American Journal of Mathematics 95 (1973), 204-214. 5. Iwasawa, K., "On the Theory of Cyclotomic Fields," Annals of Mathematics 70 (1959), 530-561. 6. Iwasawa, K., "On the µ-invariants of rz.£ -extensions, 11 Number Theory, Algebraic Geometry and Commutative Algebra, in honor of Y. Akizuki, Kinokuniya, Tokyo, 1973, 1-11. 7. Iwasawa, K., "On 'lf£-extensions of Algebraic Number Fields," Annals of Mathematics 98 (1973) 246-326. 8. Johnson, w., "Irregular Primes and Cyclotomic Invariants," Mathematics of Computation 29 (1975), 113-120. 9. Lang, s., Algebra, Addison-Wesley, 1971. 10. Serre, J.-P., "Classes des Corps Cyclotomiques," Seminaire Bourbaki, Expose 174 (1958-59).