Farshid Hajir

Christian Maire

Ravi Ramakrishna

ON OZAKI'S THEOREM REALIZING PRESCRIBED p-GROUPS AS p-CLASS TOWER GROUPS by

Keywords: 2000 Mathematics Subject Classification, -11R29, Key words and phrases, -Hilbert class field tower, p-group

We give a streamlined and effective proof of Ozaki's theorem that any finite p-group Γ is the Galois group of the p-Hilbert class field tower of some number field F. Our work is inspired by Ozaki's and applies in broader circumstances. While his theorem is in the totally complex setting, we obtain the result in any mixed signature setting for which there exists a number field k 0 with class number prime to p. We construct F{k 0 by a sequence of Z{p-extensions ramified only at finite tame primes and also give explicit bounds on rF : k 0 s and the number of ramified primes of F{k 0 in terms of #Γ.

Introduction

For a number field k, define L p pkq to be the compositum of all finite unramified Galois p-extensions of k. The extension L p pkq{k is called the p-Hilbert class field tower of k, and its Galois group GalpL p pkq{kq is its p-class tower group. In [START_REF] Ozaki | Construction of maximal unramified p-extensions with prescribed Galois groups[END_REF], Ozaki proved that every finite p-group Γ occurs as the p-class tower group of some totally complex number field F. His strategy is as follows. As finite p-groups are solvable, it is natural to proceed by induction. After establishing the base case (realizing Z{p as a p-class tower group), it remains to show that given any short exact sequence of finite p-groups [START_REF] Curtis | Representation theory of finite groups and associative algebras[END_REF] 1

Ñ Z{p Ñ G 1 Ñ G Ñ 1
where G :" GalpL p pkq{kq, one can realize G 1 as GalpL p pk 1 q{k 1 q for some number field k 1 .

Ozaki constructs such a k 1 {k via a sequence of carefully chosen Z{p-extensions.

In this paper, we provide a streamlined and effective proof of Ozaki's theorem. Some differences between our work and Ozaki's are:

-He must start with a totally complex k 0 and then construct a field F{k 0 whose p-Hilbert class field tower has the given Γ as its Galois group, while we start with a number field k 0 of arbitrary signature whose class number is prime to p.

-Our result is effective and we are able to obtain explicit upper bounds on rF : k 0 s and the number of ramified primes in F{k 0 , all of which are tame and finite. -Moreover, we bypass some of the most delicate and involved arguments of [START_REF] Ozaki | Construction of maximal unramified p-extensions with prescribed Galois groups[END_REF]. We prove:

Theorem. -Let Γ be a finite p-group and k 0 a number field with p#Cl k 0 , pq " 1. There exist infinitely many number fields F{k 0 such that GalpL p pFq{Fq » Γ and -if µ p Ć k 0 then F{k 0 is of degree at most p 2 ¨#Γ and is ramified at at most 2 `2 log p p#Γq finite tame primes, -if µ p Ă k 0 then F{k 0 is of degree at most p ¨p#Γq 2 and is ramified at at most 1 `3 log p p#Γq finite tame primes.

As any countably generated pro-p group Γ is the inverse limit of finite p-groups, Ozaki shows any such Γ is the Galois group of the maximal unramified p-extension of some infinite algebraic extension of Q. The corresponding corollary of our theorem is:

Corollary. -Any countably generated pro-p group Γ, including p-adic analytic Γ, can be realized as the p-class tower group of a totally real tamely ramified infinite extension F{Q.

We now give details about the structure of our proof and the difference between our methods and Ozaki's, though we were very much inspired by Ozaki's beautiful theorem and techniques.

We start the base case of the inductive process with any number field k 0 , of any signature, whose class number is prime to p. Referring to the group extension [START_REF] Curtis | Representation theory of finite groups and associative algebras[END_REF] with G being trivial, one has to find an extension k 1 {k 0 such that k 1 has p-class group tower exactly Z{p, which is equivalent to the p-class group being Z{p. This is a standard argument and is part of Proposition 2.8.

The base case being done, we proceed to the inductive step (with our base field relabeled k). There are two cases, depending on whether (1) splits or not. For the sake of brevity, we only outline the nonsplit case in this introduction; the split case is handled similarly. For a set of places of k, we say that an extension k 1 {k is exactly ramified at S if it is ramified at all the places in S and nowhere else. We need to find a suitable tame prime v 1 of k such that v 1 splits completely in L p pkq{k, -There is no Z{p-extension of k exactly ramified at v 1 , -The maximal p-extension L p pkq tv 1 u {L p pkq exactly ramified at the primes of L p pkq above v 1 is of degree p and solves the embedding problem [START_REF] Curtis | Representation theory of finite groups and associative algebras[END_REF]. Arranging this and its split analog are the main technical difficulties. One then chooses a second prime v 2 that also solves the embedding problem as above and remains prime in L p pkq tv 1 u {L p pkq. The existence of v 1 and v 2 will follow from Chebotarev's Theorem. The compositum of these two solutions, after a Z{p-base change k 1 {k ramified at both v 1 and v 2 (which exists!), gives the unramified solution to the embedding problem (1) which we show is L p pk 1 q. This is done in the proof of Theorem 2.

Our ability to choose primes v i as above depends upon the existence of Minkowski units in the tower L p pkq{k, namely on the condition that O Lppkq b F p » F p rGs λ ' N where N is an F p rGs-torsion module and λ is a large enough integer. In some situations, Minkowski units are rare -see §5.3 of [START_REF] Hajir | Deficiency of p-class tower groups and Minkowski units[END_REF]. By contrast, both for Ozaki's proof (implicitly) and ours (explicitly), much of the work involves seeking fields for which they exist in abundance.

If µ p Ă k, we may not be able to make our choices of v i as above to both split completely in L p pkq{k and solve the nonsplit embedding problem [START_REF] Curtis | Representation theory of finite groups and associative algebras[END_REF]. In this case we need to perform an extra base change k{k to shift the obstruction to the embedding problem so that we can proceed as above. The base change k{k must preserve the tower, that is L p p kq " L p pkq k. Theorem 1 provides such a k.

Finally we check that the condition 'λ is large enough' persists, that is there are enough Minkowski units to keep the induction going. Proposition 2.8 guarantees this. To sum up, the key ingredients of the proof of the above Theorem and Corollary are Theorems 1 and 2 and Proposition 2.8.

We now explain in some detail Ozaki's approach and our simplifications.

-Using a result of Horie, [START_REF] Horie | A note on basic Iwasawa λ-invariants of imaginary quadratic fields[END_REF], Ozaki starts with a quadratic imaginary field with class number prime to p in which p is inert. He then chooses a suitable layer k in the cyclotomic Z p -extension as the starting point of his induction. Assuming the problem solved for G in (1) and relabelling k as his base field, he proceeds inductively with the goal to find a k 1 Ą k whose p-Hilbert class field tower has Galois group G 1 . For the induction to go forward, Ozaki needs r 2 pkq ě B p pkq (implicit in this inequality is the existence of enough Minkowski units) where B p pkq is a certain explicit quantity depending on k, G and the p-part of the class group of K :" L p pkqpµ p q. This involves delicate estimates in §4 of [START_REF] Ozaki | Construction of maximal unramified p-extensions with prescribed Galois groups[END_REF]. We replace r 2 pkq ě B p pkq with f pkq ě 2h 1 pGq `3 where h i pGq :" dim H i pG, Z{pq and f pkq, which is a lower bound for the number of Minkowski units in L p pkq{k, depends only on h 1 pGq, h 2 pGq and the signature of k. We neither consider K nor invoke the estimates of §4 of [START_REF] Ozaki | Construction of maximal unramified p-extensions with prescribed Galois groups[END_REF]. -In §5 of [START_REF] Ozaki | Construction of maximal unramified p-extensions with prescribed Galois groups[END_REF], Ozaki proves his base change Proposition 1, namely he shows there exists a ramified Z{p-extension k{k such that GalpL p p kq{ kq » GalpL p pkq{kq. He uses this repeatedly when solving each embedding problem [START_REF] Curtis | Representation theory of finite groups and associative algebras[END_REF]. Several tame primes are ramified in k{k and he also needs that K and K k have the same p-class group. This makes the proof significantly more involved. Theorem 1 of this paper, our version of his Proposition 1, has only one tame prime of ramification and K plays no role. We only invoke Theorem 1 when µ p Ă k. In particular, for p odd, our Corollary above makes no use of Theorem 1. -To solve the embedding problem (1), Ozaki base changes several times (to a field relabeled k) and then uses a wildly ramified Z{p-extension L{L p pkq to solve (1). After more base changes this is switched to a solution ramified at one tame prime. He then proceeds as in the description of this work using two such solutions and a base change that absorbs the ramification at both tame primes to find a k 1 such that GalpL p pk 1 q{k 1 q " G 1 . We go directly to this last step and require at most two Z{p-base changes to solve the embedding problem. This allows us to quantify explicitly both the degree and number of ramified primes of F{k.

Notations

Let p be a prime number. ' L is a number field, O L its ring of integers, O L its units and Cl L is the p-Sylow of the class group of L.

' For a finite set S of primes of L, set V L,S " tx P L ˆ, pxq " I p , x P pL v q p @v P Su.

In particular, one has the exact sequence:

1 ÝÑ O L b F p ÝÑ V L,H {pL ˆqp ÝÑ Cl L rps ÝÑ 1.
' The superscript ^indicates the Kummer dual of an object Z defined over a number field L, though we never work with the GalpLpµ p q{Lq action on Z ^.

' L S is the maximal pro-p-extension of L unramified outside S, G S :" GalpL S {Lq and L p pLq :" L H , the maximal unramified pro-p-extension of L, as it will ease notation at various points. ' h i pHq :" dim H i pH, Z{pq. ' GovpLq :" Lpµ p qp p a V L,H q: the governing field of L. The span of tFr v u vPS in MpLq :" GalpGovpLq{Lpµ p qq controls dim H 1 pG S q.

The following may be helpful in orienting the reader:

-We frequently use finite tame primes with desired splitting properties in number field extensions. We always use Chebotarev's theorem for the existence of such primes. -Our Z{p-extensions L 1 {L of number fields are only ramified at (one or two) finite tame primes so r i pL 1 q " p ¨ri pLq and µ p Ă L 1 ðñ µ p Ă L. -Note that k 0 is our given base field, whereas k is a field used in the inductive process with p-class tower group G from (1). Our task is to construct k 1 with p-class tower group G 1 . Finally, k{k is an extension having p-class tower group G, the same as for k.

Tools for the proof

2.1. F p rGs-modules and Minkowski Units. -Let G be a finite group, a p-group in our situation. We record a few basic facts about finitely generated F p rGs-modules M.

See [START_REF] Curtis | Representation theory of finite groups and associative algebras[END_REF], §62.

Fact 1. -Any finitely generated F p rGs-module M is isomorphic to F p rGs λ ' N where N is a torsion F p rGs-module (every n P N is a torsion element) and where λ depends only on M.

Set T G :" ř gPG g. Denote by I G the augmentation ideal of F p rGs. For x P M set Ann G pxq :" tα P F p rGs | α ¨x " 0u. Let ts 1 , ¨¨¨, s h 1 pGq u be a system of minimal generators of G. By Nakayama's lemma and the fact that I G {I 2 G » G{G p rG, Gs, I G can be generated, as G-(right or left)-module, by the elements

x i :" s i ´1. Proposition 2.1. -With the x i as above, let M " F p rGs h 1 pGq and x " px 1 , x 2 , ¨¨¨, x h 1 pGq q P M. Then Ann G pxq " F p T G . Proof. -Ann G pxq " č i Ann G px i q " Ann G pxx i y h 1 pGq i"1 q " Ann G pI G q " F p T G . Proposition 2.2. -Let M " F p rGs λ ' N be a finitely generated F p rGs-module where N is torsion. Then T G pMq » F λ p .
Proof. -It is clear that T G pF p rGs λ q » F λ p . We now show T G pNq " 0. Let n P N so Ann G pnq ‰ 0. Note that Ann G pnq Ă F p rGs is a p-group stable under the action of the p-group G and thus has a fixed point. But it is easy to see the only fixed points of F p rGs are multiples of T G so T G P Ann G pnq as desired. Definition 1. -We say the tower L p pkq{k with Galois group G has λ Minkowski units if, as F p rGs-modules, V Lppkq,H {L p pkq ˆp " O Lppkq b F p » F p rGs λ ' N where N is an F p rGstorsion module.

2.2.

Extensions ramified at a tame set of primes. -We recall a standard formula on the number of Z{p-extensions of a number field with given tame ramification. See §11.3 of [START_REF] Koch | Galois Theory of p-extensions[END_REF] for a proof. Recall that for a field L, δpLq "

" 0 µ p Ć L 1 µ p Ă L . Proposition 2.3.
-Let L be a number field, p a prime number and S a set of tame primes of L prime to p. Then

h 1 pG L,S q :" dim H 1 pG L,S , Z{pq " dimpV L,S {L ˆpq ´r1 pLq ´r2 pLq ´δpLq `1 `ÿ vPS δpL v q.
Our v P S are always finite and have norm congruent to 1 mod p so δpL v q " 1. Fact 2 below follows immediately from Proposition 2.3 and the fact that GovpLq :" Lpµ p qp p a V L,H q is obtained by adjoining pth roots of elements of L to Lpµ p q. Fact 2. -Let S be a set of tame primes of L as above. For This result is Theorem 2.9 of [START_REF] Hajir | Deficiency of p-class tower groups and Minkowski units[END_REF], but we sketch the proof for the sake of keeping this paper self-contained.

Proof. -Set G " GalpL p pLq{Lq. We consider two "norm maps" induced by the norm map on units:

O LppLq Ñ O L . ´NG sending O LppLq b F p to O L O L X pO LppLq q p Ă O LppLq b F p ; ´N1 G : O LppLq b F p Ñ O L b F p . One easily sees N 1 G pO LppLq b F p q ։ N G pO LppLq b F p q
and this is an isomorphism provided O L X pO LppLq q p " pO L q p : in particular this is the case when µ p Ć L, see Proposition 2.8 of [START_REF] Hajir | Deficiency of p-class tower groups and Minkowski units[END_REF].

Write O LppLq bF p » F p rGs λ L 'N, where N is an F p rGs-torsion module. By Proposition 2.2 one has N G pO LppLq b F p q » F λ L p . Hence, when µ p Ć L dim ˜OL b F p N 1 G pO LppLq b F p q ¸" dimpO L b F p q ´λL .
When µ p Ă L, note that the 'difference' between the images of N G and N 1 G has p-rank at most dim

ˆOL XO ˆp LppLq pO L q p ˙ď h 1 pGq, so dim ˜OL b F p N 1 G pO LppLq q ¸ě dimpO L b F p q ´λL ´h1 pGq.
To conclude, we use the well-known equality (see [START_REF] Roquette | On class field towers[END_REF]Lemma 9]):

h 2 pGq ´h1 pGq " dim ˜OL b F p N 1 G pO LppLq b F p q ¸.

2.3.

Solving the ramified embedding problem with one tame prime. -We start with our nonsplit exact sequence:

1 ÝÑ Z{p ÝÑ G 1 ÝÑ G ÝÑ 1. (2)
given by the element 0 ‰ ε P H 2 pG, Z{pq. We assume that G " GalpL p pkq{kq. Set S " tvu where v is a finite tame prime of k. We first show the existence of a lift of G to G 1 in some k S {k for certain v of k. We call this solving the embedding problem [START_REF] Dekimpe | A seven-term exact sequence for the cohomology of a group extension[END_REF]

in k S . Recall that X 2 k,S ãÑ B k,S by Fact 4. Here X 2 k,H » H 2 pG k,H , Z{pq » H 2 pG, Z{pq. Let Inf S : H 2 pG k,H , Z{pq Ñ H 2 pG k,S
, Z{pq be the inflation map. We have the commutative diagram:

X 2 k,H Inf S / / _ h X 2 k,S _ g `kv b F p ˘^/ / B k,H f S / / / / B k,S
By Hoeschmann's criteria (see [START_REF] Neukirch | Cohomology of Number Fields, second editiion[END_REF]Chapter 3,[START_REF] Horie | A note on basic Iwasawa λ-invariants of imaginary quadratic fields[END_REF]), the embedding problem has a solution in k S if and only if Inf S pεq " 0. As L p pkq{k is unramified, Inf S pεq P X 2 k,S and as gpInf S pεqq " f S phpεqq P B k,S , the embedding problem has a solution if and only if hpεq P Kerpf S q. Set Gov S pkq :" kpµ p qp p a V k,S q. In the governing extensions kpµ p q Ă Gov S pkq Ă Govpkq, one sees that the kernel of the map f S : B k,H ։ B k,S is exactly the (unramified) decomposition group D v of the prime v. As noted in Fact 2, if w 1 , w 2 |v are two primes of kpµ p q, their Frobenii in GalpGovpkq{kpµ p qq differ by a nonzero scalar multiple. We have proved

1 Ñ Z{p Ñ G 1 Ñ G Ñ 1
of finite p-groups where G " GalpL p pkq{kq, there exists a finite tamely ramified extension k 1 {k with G 1 " GalpL p pk 1 q{k 1 q. To solve this embedding problem using Theorem 2, the tower L p pkq{k must have 2h 1 pGq Minkowski units. Proposition 2.7 below shows that if we start with enough Minkowski units, after a base change that realizes G 1 , we will be able to continue the induction. Proposition 2.6, which is only needed in the case when µ p Ă k, shows that given at least h 1 pGq Minkowski units, we can perform a base change that preserves the tower and the number of Minkowski units increases. Proposition 2.5 is a basic group theory result bounding h 1 pG 1 q and h 2 pG 1 q in terms of h 1 pGq and h 2 pGq. Furata proves a similar result in Lemma 2 of [START_REF] Furata | On class field towers and the rank of ideal class groups[END_REF].

Set H 2 pG 1 , Z{pq 1 :" Ker ´H2 pG 1 , Z{pq Res Ñ H 2 pZ{p, Z{pq ¯. Note h 2 pZ{pq " 1 so h 2 pG 1 q 1 is either h 2 pG 1 q or h 2 pG 1 q ´1 and in either case h 2 pG 1 q 1 ě h 2 pG 1 q ´1. Proposition 2.5. -Let 1 Ñ Z{p Ñ G 1 Ñ G Ñ 1 be a short exact sequence of finite p-groups. Then h 1 pG 1 q ď h 1 pGq `1 and h 2 pG 1 q ď h 1 pGq `h2 pGq `1.

Proof. -The h 1 result is clear. For the h 2 statement we have the long exact sequence (see for instance [START_REF] Dekimpe | A seven-term exact sequence for the cohomology of a group extension[END_REF])

0 Ñ H 1 pG, Z{pq Ñ H 1 pG 1 , Z{pq Ñ H 1 pZ{p, Z{pq G Ñ H 2 pG, Z{pq Ñ H 2 pG 1 , Z{pq 1 Ñ H 1 pG, H 1 pZ{p, Z{pqq. If G 1 Ñ G splits, we have 0 Ñ H 2 pG, Z{pq Ñ H 2 pG 1 , Z{pq 1 Ñ H 1 pG, H 1 pZ{p, Z{pqq so h 2 pG 1 q 1 ď
h 2 pGq `h1 pGq and since h 2 pG 1 q 1 ě h 2 pG 1 q ´1 the result follows. In the nonsplit case we have 0 Ñ H 1 pZ{p, Z{pq G Ñ H 2 pG, Z{pq Ñ H 2 pG 1 , Z{pq 1 Ñ H 1 pG, H 1 pZ{p, Z{pqq so h 2 pG 1 q 1 ď h 2 pGq ´1 `h1 pGq so h 2 pG 1 q ď h 1 pGq `h2 pGq. Definition 2. -For a number field L set G " GalpL p pLq{Lq. Define f as follows: f pLq " " r 1 pLq `r2 pLq ´h2 pGq `h1 pGq ´1 µ p Ć L r 1 pLq `r2 pLq ´h2 pGq µ p Ă L .

Fact 5 implies f pLq is a lower bound on the number of Minkowski units of L p pLq{L.

Proposition 2.6. -Let k{k be a Z{p-extension ramified at finite tame primes such that G " GalpL p pkq{kq " GalpL p p kq{ kq. Then f p kq " f pkq `pp ´1qpr 1 pkq `r2 pkqq.

Proof. -This follows immediately as we have the same group G for k and k, µ p Ă k ðñ µ p Ă k and r i p kq " p ¨ri pkq.

Proposition 2.7. -Let k 1 {k be a tamely ramified Z{p-extension such that G " GalpL p pkq{kq and G 1 " GalpL p pk 1 q{k 1 q where

1 Ñ Z{p Ñ G 1 Ñ G Ñ 1.
Let f pkq be as in Definition 2. Then f pkq ě 2h 1 pGq `3 ùñ f pk 1 q ě 2h 1 pG 1 q `3.

Proof. -We do the case µ p Ć k first. We need to prove r 1 pkq `r2 pkq ´h2 pGq `h1 pGq ´1 ě 2h 1 pGq `3 ùñ r 1 pk 1 q `r2 pk 1 q ´h2 pG 1 q `h1 pG 1 q ´1 ě 2h 1 pG 1 q `3, that is r 1 pk 1 q `r2 pk 1 q ? ě h 1 pG 1 q `h2 pG 1 q `4. Clearly r 1 pk 1 q `r2 pk 1 q " ppr 1 pkq `r2 pkqq ě pph 1 pGq `h2 pGq `4q and by Proposition 2.5 we have h 2 pG 1 q `h1 pG 1 q `4 ď ph 1 pGq `h2 pGq `1q `ph 1 pGq `1q `4 " 2h 1 pGq `h2 pGq `6 so it suffices to show pp ´1qh 2 pGq `pp ´2qh 1 pGq `4p ě 6.

This holds for all p.

When µ p Ă k. We need to prove r 1 pkq `r2 pkq ´h2 pGq ě 2h 1 pGq `3 ùñ r 1 pk 1 q `r2 pk 1 q ´h2 pG 1 q ě 2h 1 pG 1 q `3, that is r 1 pk 1 q `r2 pk 1 q ? ě 2h 1 pG 1 q `h2 pG 1 q `3. Again using Proposition 2.5 and that r i pk 1 q " p ¨ri pkq it suffices to show pp ´1qh 2 pGq `p2p ´3qh 1 pGq `3p ě 6 which holds for all p. Proposition 2.8 below provides the base case of the induction. Proposition 2.8. -Recall p#Cl k 0 , pq " 1. There exists a tamely ramified extension k 1 {k 0 such that -the p-part of the class group of k 1 is Z{p, -rk 1 : k 0 s " p 3 , -and f pk 1 q ą 2h 1 pZ{pq `3 " 5.

Proof. -Since L p pk 0 q " k 0 , we see G " teu. Choose a tame prime v of k whose Frobenius is trivial in the governing Galois group Mpkq. By Fact 2 there is a unique Z{p-extension k 1 {k 0 . That p#Cl k 1 , pq " 1 follows from Fact 3. Repeat this process with k 1 to get a field k 2 with p#Cl k 2 , pq " 1.

We do one more base change to find a field k 1 with class group Z{p. This is proved more generally as part of Theorem 2, but we include a short proof here.

Choose v 1 a finite tame prime of k 2 with trivial Frobenius in Govpk 2 q so that by Fact 2 there exists a unique D 1 {k 2 ramified at v 1 . As 1 X Govpk 2 q " k 2 , we may choose v 2 a finite tame prime of k 2 with trivial Frobenius in Govpk 2 q such that v 2 remains prime in f pk 1 q ě r 1 pk 1 q `r2 pk 1 q ´h2 pZ{pq " p 3 r 1 pkq `p3 r 2 pkq ´1 ą 5 " 2h 1 pZ{pq `3.

D
Depending on p and the signature of k 0 one can decrease the number of base changes, but this analysis complicates the statement of the main theorem without significant gain.

Solving the embedding problem

Having established the base case of our induction, we now prove Theorem 2, the main

Inductive Step. -Let 1 Ñ Z{p Ñ G 1 Ñ G Ñ 1
be exact and let k be a number field with GalpL p pkq{kq " G and f pkq ě 2h 1 pGq `3. Then there exists a number field k 1 {k with GalpL p pk 1 q{k 1 q " G 1 and f pk 1 q ě 2h 1 pG 1 q `3.

Theorem 1 below is only necessary for the key inductive step, Theorem 2, when µ p Ă k.

Set K :" L p pkqpµ p q. We only consider finite tame primes v of k that split completely in K{k. When µ p Ć k, our Frobenii in governing fields (or their subfields) are only defined up to scalar multiples. We write xFr v y Govpkq{kpµpq for the well-defined line spanned by Frobenius at v in GalpGovpkq{kpµ p qq. When the Frobenius is trivial there is no ambiguity so we write xFr v y Govpkq{kpµpq " 0.

We need primes v of k that let us control h 1 pGalpk tvu {kqq and h 1 pGalpL p pkq tvu {L p pkqqq simultaneously via Fact 2. Recall MpL p pkqq :" GalpGovpL p pkqq{L p pkqpµ p qq » F p rGs λ k 'N where N is a torsion module over F p rGs. We have no knowledge of N and must work with the free part to control things over L p pkq. We then use Proposition 3.1 to control things over k.

I G ¨MpLppkqq GovpkqK ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ K s s s s s s s s s s s MpLppkqq Govpkq L p pkq ∆ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ kpµ p q s s s s s s s s s G k G ∆ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈
.

Let ∆ " Galpkpµ p q{kq " GalpK{L p pkqq. As GalpF {Kq :" MpL p pkqq{I G ¨MpL p pkqq is the maximal quotient of MpL p pkqq on which G acts trivially, and ∆ acts on GalpF {Kq by scalars, the line xFr w y F {K is invariant under the action of GalpK{kq " G ˆ∆. Since the w|v form an orbit under this action of GalpK{kq, this line is independent of the choice of w|v as desired.

As GovpkqK{K ascends from Govpkq{kpµ p q, we see G acts trivially on GalpGovpkqK{Kq so GovpkqK Ă F . Below, we implicitly use that our primes of k split completely in K. If xFr v 1 y F {K " xFr v 2 y F {K , these lines are equal when projected to GalpGovpkqK{Kq Ă GalpGovpkqK{kpµ p qq and they are again equal in GalpGovpkq{kpµ p qq so xFr v 1 y Govpkq{kpµpq " xFr v 2 y Govpkq{kpµpq . The last statement is clear.

Theorem 1. -Recall tx i u h 1 pGq i"1
is a minimal set of generators of I G . Assume that f pkq ě h 1 pGq. Let w be a degree one prime of K such that

Fr w " ppx 1 , x 2 , ¨¨¨, x h 1 pGq , 0, ¨¨¨, 0q, 0q P MpL p pkqq » F p rGs λ k ' N.
Then for v of k below w, xFr v y Govpkq{kpµpq " 0 so there exists a Z{p-extension k{k ramified at v. Furthermore, L p p kq " L p pkq k and f p kq ą f pkq.

Proof. -As Fr w projects to 0 in the F p -vector space GalpF {Kq, Proposition 3.1 implies xFr v y Govpkq{kpµpq " 0 so k exists by Fact 2. We show the F p rGs-span of px 1 , ¨¨¨, x h 1 pGq q P F p rGs h 1 pGq has dimension #G ´1 by computing the dimension of X h 1 pGq i"1 Annpx i q. This intersection is the annihilator of I which by Proposition 2.1 is just F p T G , establishing our dimension result. By Fact 2 there is one extension over L p pkq ramified at v and thus it must be L p pkq k. Fact 3 applied to L p pkq k{L p pkq implies p#Cl Lppkq k, pq " 1 so L p p kq " L p pkq k. Proposition 2.6 gives f p kq ą f pkq.

The inductive step. -

Theorem 2. -Assume that L p pkq{k has λ k ě 2h 1 pGq `3 Minkowski units. Let 1 Ñ Z{p Ñ G 1 Ñ G Ñ 1. If µ p Ć k (resp. µ p Ă k)
there exists a field k 1 {k that is a Z{p-extension (resp. a compositum of two successive Z{p-extensions) such that GalpL p pk 1 q{k 1 q » G 1 and L p pk 1 q{k 1 has at least 2h 1 pG 1 q `3 Minkowski units.

Proof. -Recall that our finite tame primes split completely in K{k. We first treat the split case. This is independent of whether or not µ p Ă k. Split case. Choose tame degree one primes w 1 and w 2 GovpkqK such that -Fr w 1 " ppx 1 , x 2 , ¨¨¨, x h 1 pGq , 0, ¨¨¨, 0q, 0q P GalpGovpL p pkqq{GovpkqKq Ă MpL p pkqq. This is possible as the tuple lies in I G ¨MpL p pkqq and GovpkqK Ă F . As Fr w 1 projects to 0 in GalpF {Kq, we see for v 1 of k below w 1 that xFr v 1 y F {K " 0 so by Proposition 3.1 xFr v 1 y Govpkq{kpµpq " 0. By Fact 2 applied to k there is one Z{pextension D 1 {k ramified at v 1 . Fact 2 also gives (see the proof of Theorem 1 as well) a unique Z{p-extension of L p pkq ramified at v 1 , namely D 1 L p pkq{L p pkq. -Fr w 2 " pp0, 0, ¨¨¨, 0 h 1 pGq , x 1 , x 2 , ¨¨¨, x h 1 pGq , 0, 0, 0, ¨¨¨0q, 0q so for v 2 of k below w 2 , xFr v 2 y F {K " 0. We also insist that v 2 remains prime in D 1 {k. This last condition is linearly disjoint from the rest of the defining splitting conditions on v 2 and imposes no contradiction. Again, there is one Z{p-extension of both k and L p pkq ramified at v 2 , namely D 2 {k. Let D{k be a 'diagonal' extension between D 1 and D 2 ramified at both v 1 and v 2 . There are p ´1 of these. Fact 2 and our choices of the Frobenii of v 1 and v 2 imply h 1 `GalpL p pkq tv 1 ,v 2 u {L p pkq ˘" 2. (With only h 1 pGq Minkowski units, we would have had h 1 `GalpL p pkq tv 1 u {L p pkq ˘" h 1 `GalpL p pkq tv 2 u {L p pkq ˘" 1, but h 1 `GalpL p pkq tv 1 ,v 2 u {L p pkq ˘ą 2.)

J p,el v 2 "? • • • • • • • • • • • • • • • • Π L :" D 1 D 2 L p pkq H ? ✤ ✤ ✤ ✤ ✤ ✤ E 0 ? q q q q q q ▼ ▼ ▼ ▼ ▼ ▼ ▼ D 2 L p pkq ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ DL p pkq J :" D 1 L p pkq | | | | | | | | | | | | | D 2 D D 1 L p pkq Ω"Z{p ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ Govpkq ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ k ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❳❳ ❳❳ ❳❳ ❳❳ ❳❳ ❳❳ ❳❳ ❳❳ ❳❳ ❳❳ ❳❳ ❳❳ ❳❳ ❳❳ ❳❳ ❳❳ ❳❳ ❳❳ ❳❳ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣
Set L :" D 1 D 2 L p pkq, J :" D 1 L p pkq and note L{D is unramified as D{k has absorbed all ramification at tv 1 , v 2 u. We will solve the problem by showing p#Cl D 1 D 2 Lppkq , pq " 1. Since p#Cl Lppkq , pq " 1 and our choice of v 1 is such that h 1 pGalpL p pkq tv 1 u {L p pkqq " 1, Fact 3 applied to J{L p pkq implies p#Cl J , pq " 1.

We now prove that there exists a unique Z{p-extension over J unramified outside v 2 , namely L. Set Ω " GalpJ{L p pkqq, J p,el tv 2 u to be the maximal elementary p-abelian extension of J inside J tv 2 u , and Π " GalpJ p,el tv 2 u {Jq. Then Ω acts on Π and trivially on GalpL{Jq. We claim this is the only Z{p-extension of J in J p,el tv 2 u {J on which Ω acts trivially: If not, there exists another Z{p-extension H{J unramified outside v 2 and Galois over L p pkq. Hence GalpH{L p pkqq has order p 2 and is abelian. The extension H{L p pkq cannot be cyclic because all inertia elements have order p and would then fix an everywhere unramified extension of L p pkq, a contradiction. Suppose now that GalpH{L p pkqq » Z{p ˆZ{p, with H ‰ JD 2 " L. Then GalpHD 2 {L p pkqq » pZ{pq 3 : this contradicts the already established fact that h 1 pGalpL p pkq tv 1 ,v 2 u {L p pkqq " 2. The final possibility is that there exists a Z{p-extension E 0 {J unramified outside v 2 , different from L{J and not fixed by Ω; let S 0 be the set of ramification of E 0 {J. As primes above v 2 in L p pkq are inert in J{L p pkq, ΩpS 0 q " S 0 : then Ω takes E 0 to another Z{p-extension E 1 {J exactly ramified at S 0 and such that E 1 ‰ E 0 . The compositum E 1 E 0 {J contains a Z{p-extension E 1 0 {J exactly ramified at a set S 1 0 Ĺ S 0 . Observe that E 1 0 ‰ L since L{J is totally ramified at every prime above v 2 . Continuing the process, we obtain an unramified Z{p-extension H{J, which is impossible since p#Cl J , pq " 1. Thus L{J is the unique Z{p-extension unramified outside v 2 . Fact 3 applied to L{J implies p#Cl L , pq " 1. We have solved the split embedding problem with k 1 " D and GalpL p pk 1 q{k 1 q " G ˆZ{p. It required one base change ramified at two tame finite primes. Proposition 2.7 implies f pk 1 q ě 2h 1 pG 1 q `3 so the induction can proceed.

For the nonsplit case we treat µ p Ć k and µ p Ă k separately. Theorem 1 is only used in the nonsplit case when µ p Ă k.

The nonsplit case, µ p Ć k. By Lemma 2.4 we may use one tame prime v of k to find a ramified solution to the embedding problem. As µ p Ć k implies Govpkq X L p pkq " k, we can assume v splits completely in K{k. Choosing any w|v of K we set Fr w " ppz 1 , z 2 , ¨¨¨, z λ k q, n 0 q P MpL p pkqq where we claim n 0 R I G ¨N and z i P I G Ă F p rGs. Indeed, if any z i R I G , its F p rGs-span is all of F p rGs and by Fact 2 there is no Z{pextension of L p pkq ramified at the w|v, contradicting that we are solving an embedding problem with v. If n 0 P I G ¨N, then the projection of Fr w to GalpF {Kq is trivial so Proposition 3.1 implies xFr v y Govpkq{kpµpq " 0 and the embedding problem we are solving is split, also a contradiction. Choose a degree one w 1 of K with Fr w 1 " ppx 1 , x 2 , ¨¨¨, x h 1 pGq , 0, 0, 0, ¨¨¨0q, n 0 q P MpL p pkqq where n 0 is as in the previous paragraph. Let v 1 be the prime of k below w 1 . By Fact 2 (also see the proof of Theorem 1) there is one Z{p-extension D 1 {L p pkq ramified at v 1 . Choose a degree one w 2 of K with Fr w 2 " pp0, 0, ¨¨¨, 0, x 1 , x 2 , ¨¨¨, x h 1 pGq , 0, 0, 0, ¨¨¨0q, n 0 q P MpL p pkqq and the primes of L p pkq above v 2 remain prime in D 1 {L p pkq. This last condition is linearly disjoint from the splitting conditions defining v 2 and imposes no contradiction. Again by Fact 2 there is one Z{p-extension D 2 {L p pkq ramified at v 2 .

As the free components of of Fr w , Fr w 1 and Fr w 2 are all in I λ k G , their projections to GalpF {Kq depend only on n 0 and Proposition 3.1 implies 0 ‰ xFr v y Govpkq{k " xFr v 1 y Govpkq{k " xFr v 2 y Govpkq{k . Thus there is no extension of k ramified at either v 1 or v 2 , but, by Fact 2, there is a Z{p-extension of k ramified at tv

1 , v 2 u. Call it D. Note D 1 • • • • • • • • • • • • • • • • D 2 DL p pkq L p pkq • • • • • • • • • • • • • • • | | | | | | | | | | | | | | | | | D G k G • • • • • • • • • • • • • • • • • • • •
That D 1 D 2 has trivial p-class group follows exactly as it did in the split case and we may set k 1 " D so L p pk 1 q " D 1 D 2 and GalpL p pk 1 q{k 1 q » G 1 . We have solved the embedding problem in the nonsplit case when µ p Ć k. We performed one base change ramified at two tame finite primes and Proposition 2.7 implies f pk 1 q ě 2h 1 pG 1 q `3 so the induction can proceed.

The nonsplit case, µ p Ă k. We can no longer assume L p pkq X Govpkq " k. Let 0 ‰ ε P X 2 k,H be the obstruction to our embedding problem G 1 ։ G. Using Lemma 2.4, let v of k be a tame prime annihilating ε. The difficulty is that in the diagram below we may have L p pkq X Govpkq Ľ k and that Fr v , which is necessarily nonzero in Mpkq, may also be nonzero in GalppL p pkq X Govpkqq{kq. This prevents us from also choosing v to split completely in L p pkq{k and as we need in GovpL p pkqq{L p pkq to ensure there is only one extension of L p pkq ramified at the primes of L p pkq above v. If we could choose v to annihilate ε such that Fr v " 0 P GalpL p pkq{kq, we would be able to proceed as in the µ p Ć k case. We get around this by a base change. By Kummer theory and the definition of governing fields, GalpGovpLq{Lpµ p qq is an elementary p-abelian group. Let k{k be a tamely ramified Z{p-extension as given by Theorem 1 so GalpL p p kq{ kq " G. By Proposition 2.6 we have λk ě 2h 1 pGq `3.

L p p kq Govp kq

L p pkq L p p kq X Govp kq

♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ L p pkq X Govpkq k k ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ Z{p pZ{pq r
As Govpkq X k " k, we may choose a prime v to solve the embedding problem for k whose Frobenius is nontrivial in Galp k{kq, that is v remains prime in k{k. As observed above, L p p kqXGovp kq{ k is a pZ{pq r -extension for some r and, as GalpL p pkq{kq " GalpL p p kq{ kq " G, it is the base change of such a subextension of L p pkq{k from k so L p p kq X Govp kq{k is a pZ{pq r`1 -extension. Since v remains prime in k{k and residue field extensions are cyclic, it splits completely in L p p kq X Govp kq{ k. As the embedding problem is solvable over k by allowing ramification at v, it is also solvable over k by allowing ramification at the unique prime of k above v. Thus ε P X 2 k,H ãÑ Bk ,H " Mp kq actually lies in Gal `Govp kq{ `Lp p kq X Govp kq ˘˘. The base change shifted the obstruction to outside of our p-Hilbert class field tower! The rest of the proof is identical to the µ p Ć k case.

We now prove the Main Theorem of the Introduction:

Proof. -We have verified the base case of the induction in Proposition 2.8 and the inductive step with Theorem 2. It remains to count degrees and ramified primes. Proposition 2.8 involved three Z{p-base changes, the first two ramified at one tame prime and the last at two tame primes. The inductive steps breaks into cases as follows µ p Ć k 0 : At each of the log p p#Γq ´1 inductive stages we need one base change ramified at two primes for a total of 3 ``log p p#Γq ´1˘b ase changes ramified at 4 `2plog p p#Γq ´1q primes.

µ p Ă k 0 : At each of the log p p#Γq ´1 inductive stages we need two base changes, the first ramified at one prime and the second at two primes. There are 3 2 `log p p#Γq ´1˘b ase changes ramified at 4 `3plog p p#Γq ´1q primes.

Fact 4 .

 4 -X 2 L,S ãÑ B L,S . Let λ L be the number of Minkowski units in L p pLq{L. Fact 5. -If µ p Ć L then λ L " r 1 pLq `r2 pLq ´1 `h1 pGq ´h2 pGq. If µ p Ă L then λ L ě r 1 pLq `r2 pLq ´h2 pGq.

Lemma 2 . 4 . 2 . 4 .

 2424 -The embedding problem (2) has a solution in k S {k if and only if hpεq P D v . Thus it has a solution in k S {k if we choose the prime v such that xFr v y " xhpεqy in Mpkq, that is the lines spanned by these elements in Mpkq are equal. This is always possible by Chebotarev's Theorem. Cohomological facts implying the persistence of Minkowski units. -Our main aim in this paper is to show that given a short exact sequence

  {L ˆpq ^. Recall X 2 L,S :" Ker pH 2 pG S , Z{pq Ñ ' vPS H 2 pG v , Z{pqq. Fact 4 below is well-known. See Theorem 11.3 of[START_REF] Koch | Galois Theory of p-extensions[END_REF].

each v P S let Fr v P MpLq :" GalpGovpLq{Lpµ p qq. If the set tFr v , v P Su spans an p#S ´dq-dimensional subspace of MpLq, then dim H 1 pG L,S , Z{pq " d `dim H 1 pG L,H , Z{pq. When µ p Ć k, Fr v is only well-defined up to nonzero scalar multiplication. Fact 3. -Let L be a number field such that p#Cl L , pq " 1. Let L 1 {L be a Z{p-extension exactly ramified at S " tv 1 , ¨¨¨, v r u where the v i are finite and tame. Then p#Cl L 1 , pq " 1 if and only if L 1 {L is the unique Z{p-extension of L unramified outside S. In particular, that is the case when |S| " 1.

Proof. -Indeed, p#Cl L 1 , pq ‰ 1 if and only if there exists an unramified Z{p-extension H{L 1 such that H{L is Galois (use the fact the the action of a p-group on a p-group always has fixed points). Observe that H{L cannot be cyclic of degree p 2 as all inertial elements of GalpH{Lq have order p and they would thus fix an unramified extension of L, a contradiction. So GalpH{Lq » Z{pˆZ{p, and L has at least two disjoints Z{p-extension unramified outside S, also a contradiction.

Set B L,S " pV L,S

  1 {k 2 . Again by Fact 2 there exists a unique D 2 {k 2 ramified at v 2 . Let D{k 2 be any of the p ´1 'diagonal' Z{p-extensions of k 2 between D 1 and D 2 so D 1 D 2 {D is everywhere unramified. We claim D 1 D 2 " L p pDq. Indeed, by Fact 3 applied to D 1 {k 2 we see p#Cl D 1 , pq " 1. As v 2 is inert in D 1 {k 2 , the extension D 2 D 1 {D 1 is ramified only at v 2 and Fact 3 applied to D 2 D 1 {D 1 implies p#Cl D 1 D

2 , pq " 1. Whether or not µ p Ă k, we have k 1 :" D, Cl k 1 " Z{p and

3.1. The Stability Theorem. - Proposition 3.1

  . -Let F Ă GovpL p pkqq be the field fixed by I G ¨MpL p pkqq. For v of k splitting completely in K and w|v in K, the lines xFr w y F {K do not dependent on w so we may write xFr v y F {K . Then xFr v 1 y F {K " xFr v 2 y F {K implies xFr v 1 y Govpkq{kpµpq " xFr v 2 y Govpkq{kpµpq . If xFr v 1 y F {K " 0 then xFr v 1 y Govpkq{kpµpq " 0.

	Proof. -This diagram is useful in Theorems 1 and 2 as well.
		GovpL p pkqq
	F	s s s s s s s s s s

The second author was partially supported by the ANR project FLAIR (ANR-17-CE40-0012) and by the EIPHI Graduate School (ANR-17-EURE-0002). The third author was partially supported by Simons Collaboration grant #524863. He also thanks FEMTO-ST for its hospitality and wonderful research environment during his visit there in the spring of 2022. All three authors were supported for a Collaborate@ICERM visit in January, 2022.