The m-step solvable anabelian geometry of number fields

Mohamed Saïdi and Akio Tamagawa

Abstract

Given a number field K and an integer m0, let Km denote the maximal m-step solvable Galois extension of K and write GKm for the maximal m-step solvable Galois group Gal(Km/K) of K. In this paper, we prove that the isomorphy type of K is determined by the isomorphy type of GK3. Further, we prove that Km/K is determined functorially by GKm+3 (resp. GKm+4) for m2 (resp. m1). This is a substantial sharpening of a famous theorem of Neukirch and Uchida. A key step in our proof is the establishment of the so-called local theory, which in our context characterises group-theoretically the set of decomposition groups (at nonarchimedean primes) in GKm, starting from GKm+2.

0 Introduction/main results

Let K and L be number fields with algebraic closures K¯ and L¯ and absolute Galois groups

GK=defGal(K¯/K)andGL=defGal(L¯/L),

respectively. A celebrated theorem of Neukirch and Uchida states that (profinite) group isomorphisms between GK and GL arise functorially from field isomorphisms between K and L. More precisely, one has the following (cf. [8, 12]), which is the first established (birational) anabelian result of this sort (well before Grothendieck announced his anabelian programme).

Theorem (Neukirch–Uchida theorem).

Let τ:GKGL be an isomorphism of profinite groups. Then there exists a unique field isomorphism σ:K¯L¯ such that τ(g)=σgσ-1 for every gGK. In particular, σ(K)=L and K, L are isomorphic.

Moreover, the theorem is still valid when one replaces GK, GL by their respective maximal prosolvable quotients GKsol, GLsol and K¯, L¯ by the maximal prosolvable extensions Ksol, Lsol of K in K¯ and L in L¯, respectively (cf. [7, 13]). Thus, the isomorphy type of the (maximal prosolvable) Galois group of a number field determines functorially the isomorphy type of the number field. An explicit description of the isomorphy type of the (prosolvable) Galois group of a number field seems to be out of reach for the time being. This prompts the natural question:

Question.

Is it possible to prove any refinement of the Neukirch–Uchida theorem, whereby one replaces the full (prosolvable) Galois groups of number fields by some profinite quotients whose structure can be better approached/understood?

Class field theory provides a description of the maximal abelian quotient of the Galois group of a number field. The structure of the maximal m-step solvable quotient of the Galois group of a number field can be in principle approached via class field theory, say for small values of m as in (commutative) Iwasawa theory in case m=2. In this paper we prove the following sharpening of the Neukirch–Uchida theorem. Write GKm and GLm for the maximal m-step solvable quotients of GK, GL and Km/K, Lm/L for the corresponding subextensions of K¯/K and L¯/L, respectively (cf. Notations).

Theorem 1.

Assume there exists an isomorphism τ3:GK3GL3 of profinite groups. Then there exists a field isomorphism σ:KL.

Theorem 2.

Let m0 be an integer and τm+3:GKm+3GLm+3 an isomorphism of profinite groups.

1. There exists a field isomorphism σm:KmLm such that τm(g)=σmgσm-1 for every gGKm, where τm:GKmGLm is the isomorphism induced by τm+3. In particular, σm induces an isomorphism KL.

2. Assume m2 (resp. m=1). Then the isomorphism σm:KmLm (resp. σ:KL induced by σ1:K1L1) in assertion (i) is uniquely determined by the property that τm(g)=σmgσm-1 for every gGKm, where τm:GKmGLm is the isomorphism induced by τm+3.

Theorem 1 is not functorial in the sense that we do not know a priori how an isomorphism σ in the underlying statement relates to the isomorphism τ3. In this respect Theorem 1 is a sharpening of the weak version of the Neukirch–Uchida theorem originally proved in [8]. The uniformity assertion in the Neukirch–Uchida theorem (existence and uniqueness of σ therein) was established by Uchida in [12, 13], using among others Neukirch’s results. Uchida also removed the assumption originally imposed by Neukirch that at least one of K and L is Galois over the prime field .

Theorem 2 above is functorial. It implies in particular that the isomorphy type of K is functorially determined by the isomorphy type of GK4.

As in the proof of the Neukirch–Uchida theorem, a key step in the proofs of Theorems 1 and 2 is the establishment of the so-called local theory, i.e. starting from an isomorphism of (quotients of) absolute Galois groups of number fields one establishes a one-to-one correspondence between the sets of their nonarchimedean primes and one between the corresponding decomposition groups. The latter is usually achieved via a purely group-theoretic characterisation of decomposition groups. (In the proof of the Neukirch–Uchida theorem decomposition groups are characterised group-theoretically using the method of Brauer groups.) In our context we prove the following (cf. Theorem 25 and Corollary 27 (i)).

Theorem 3.

Let m2 (resp. m=1) be an integer. Then one can reconstruct group-theoretically the set of nonarchimedean primes of Km (resp. K) and the set of decomposition groups of GKm at those primes starting from the profinite group GKm+2.

Before proving Theorem 3, we establish a certain separatedness result that enables one to recover the set of nonarchimedean primes of Km once one has recovered the set of decomposition groups in GKm. More precisely, we prove that the natural surjective map from the set of nonarchimedean primes of Km to the set of decomposition groups in GKm is bijective if m2 (cf. Corollary 6).

To conclude the proof of Theorem 1, we use Theorem 3 and resort to a recent result of Cornelissen, de Smit, Li, Marcolli and Smit in [2]. In order to apply this result, one needs, in addition to recovering the decomposition groups in GK1 starting from GK3 (as in Theorem 3), to recover the Frobenius elements in these decomposition groups (modulo inertia groups). One of the key technical results we establish to that effect is that one can recover group-theoretically the cyclotomic character of GK1 starting from GK3 (cf. Theorem 26).

Concerning the proof of Theorem 2. Once the above local theory in our context is established (cf. Theorem 3), the rest of the proof of Theorem 2 is somewhat similar to that of the Neukirch–Uchida theorem with some necessary adjustments. It does not rely on any of the results in [2] mentioned above.

In Appendix A, we establish the result that, for every prime number l, one can recover group-theoretically the l-part of the cyclotomic character of GK up to twists by finite characters starting from GK2 (cf. Proposition 9). This result is optimal as it is not possible to obtain a similar result starting solely from GK1. Indeed, it is not possible in general to distinguish (group-theoretically) the l-quotient of GK1 corresponding to the cyclotomic l-extension of K among the various l-quotients of GK1. This phenomenon seems to be one of the main reasons why the isomorphy type of K is not encoded in the isomorphy type of GK1 as is well known (see [1], for example). In order to prove this result we use the machinery and techniques of Iwasawa theory. Our proof of recovering the l-part of the cyclotomic character (up to twists by finite characters) relies on a careful analysis of the structure of annihilators of certain Λ-modules, where Λ is the (multi-variable in general) Iwasawa algebra associated to the maximal pro-l abelian torsion free quotient of GK.

Our results are almost optimal. As mentioned above, Theorem 1 does not hold if one starts with an isomorphism τ1:GK1GL1, as is well known. The best improvements of Theorems 1 and 2 one can hope for are the following.

Question 1.

Does the conclusion in Theorem 1 hold if one starts with an isomorphism τ2:GK2GL2?

Question 2.

Can one replace the m+3 in Theorem 2 by m+i, for some 0i<3?

Relation to other works

The main result in [2] (mentioned above) states that two global fields are isomorphic if and only if their abelianised Galois groups are isomorphic and some extra (a priori non-Galois-theoretic) conditions hold (cf. [2, Theorem 3.1]). Our Theorem 3, together with the fact established in Theorem 26 that one can recover the cyclotomic character from GK3, implies that the information of K needed to apply the main theorem of [2] is group-theoretically encoded in GK3. The proof of our Theorem 1 follows then by applying the main theorem of [2]. It would be interesting to investigate how precisely the conditions in the main theorem of [2] relate to the Galois-theoretic information of K and how precisely this main theorem relates to our Theorem 1. Also, contrary to the proof of Theorem 1, our proof of Theorem 2 does not rely on the main (or any other) result in [2]. In particular, Theorem 2 (i) for m=0 gives an alternative proof of Theorem 1.

Our main Theorems 1 and 2 are bi-anabelian, i.e. with a reference to two, a priori distinct, number fields. A mono-anabelian version of Theorem 2 would be a version whereby one establishes a purely group-theoretic algorithm which starting solely from an isomorph of GKn, for suitable n1, reconstructs an isomorph of the number field K. In [4] Hoshi establishes such an algorithm starting from an isomorph of GK. More generally, he also establishes such an algorithm starting from an isomorph of the maximal prosolvable quotient GKsol of GK, under the assumption that the maximal prosolvable extension Ksol of K in K¯ is Galois over the prime field . The algorithm relies crucially on this assumption, as well as the Neukirch–Uchida theorem itself. In fact, Hoshi’s algorithm does not provide an alternative proof of the Neukirch–Uchida theorem but rather uses it in an essential way. In our context, one could adapt Hoshi’s arguments to show, using Theorems 2 and 3, the following (details of proof may be considered in a subsequent work).

Let m0 be an integer, and assume that Km is Galois over Q. Then there exists a purely group-theoretic algorithm which starting from GKn, for a suitable integer nm that can be made effective, reconstructs functorially the field Km together with the natural action of GKm on Km.

Finally, we mention that the authors prove an (a mono-anabelian) analogue of the main results of this paper for global function fields in positive characteristics (cf. [11]).

Future perspectives

The Neukirch–Uchida theorem had several deep and substantial applications in anabelian geometry and has played a prominent role in the theory for more than 40 years. We hope that Theorem 2 will have a similar impact, for example in developing a new anabelian geometry, over finitely generated fields, stemming solely from the arithmetic of m-step solvable extensions of finitely generated fields and m-step solvable arithmetic fundamental groups, for small values of m.

Notations

Given a finite set H, we write |H| for its cardinality.

For a profinite group G let [G,G]¯ be the closed subgroup of G which is (topologically) generated by the commutator subgroup of G. We write Gab=defG/[G,G]¯ for the maximal abelian quotient of G.

Given a profinite group G, and a prime number l, we write G(l) for the maximal pro-l quotient of G, and G(l) for the maximal prime-to-l quotient of G.

Let G be a profinite group and consider the derived series

G[i+1]G[i]G[1]G[0]=G,

where G[i+1]=[G[i],G[i]]¯, for i0, is the (i+1)-st derived subgroup which is a characteristic subgroup of G. We write Gi=defG/G[i] and refer to it as the (maximal) i-step solvable quotient of G (thus G1=Gab, G2 is the maximal metabelian quotient of G, etc.). By definition, G[i]=Ker(GGi). For ji0 we write

G[j,i]=defKer(GjGi)=Gj[i]=G[i]j-i.

We write

Gsol=defG/(i0G[i])=limi0Gi

and refer to it as the (maximal) prosolvable quotient of G.

Given a profinite group G we write Sub(G) for the set of closed subgroups of G, and C(G) for the centre of G. For HSub(G), we write NG(H) for the normaliser of H in G.

Given a profinite group G and a prime number l, we write Gl for an l-Sylow subgroup of G, which is defined up to conjugation.

Let G be a profinite group, HG a closed subgroup, and l a prime number. We say that H is l-open in G if an l-Sylow subgroup of H is open in an l-Sylow subgroup of G. We say that G is l-infinite if an l-Sylow subgroup of G is infinite, or, equivalently, if {1} is not l-open in G. We say that two subgroups H1,H2G are commensurable (resp. l-commensurable) if H1H2 is open (resp. l-open) in both H1 and H2. The intersection of two open (resp. l-open) subgroups of G is open (resp. is not l-open in general). Accordingly, commensurability (resp. l-commensurability) relation is (resp. is not in general) an equivalence relation on Sub(G).

Given an abelian profinite group A, we write Ator¯ for the closure in A of the torsion subgroup Ator of A, and set A/tor=defA/Ator¯. Given a profinite group G, we set

Gab/tor=def(Gab)/tor.

Given a field K, we write K¯ for an algebraic closure of K, Ksep for the maximal separable extension of K contained in K¯, and GK for the absolute Galois group Gal(Ksep/K) of K.

Given a field K and an integer m0, we write Km/K for the maximal m-step solvable subextension of Ksep/K, which corresponds to the quotient GKGKm. By definition, we have GKm=GK[m].

Given a field K, and HAut(K) a group of automorphisms of K, we write KHK for the subfield of K which is fixed under the action of H.

A number field is a finite field extension of the field of rational numbers . For an (possibly infinite) algebraic extension F of , we write 𝔓𝔯𝔦𝔪𝔢𝔰F (resp. 𝔓𝔯𝔦𝔪𝔢𝔰Fna) for the set of primes (resp. nonarchimedean primes) of F. We often identify 𝔓𝔯𝔦𝔪𝔢𝔰na with the set of prime numbers. For FF¯ and 𝔭𝔓𝔯𝔦𝔪𝔢𝔰Fna, we write 𝔭F𝔓𝔯𝔦𝔪𝔢𝔰Fna for the image of 𝔭 in 𝔓𝔯𝔦𝔪𝔢𝔰Fna. Further, for a set of primes S𝔓𝔯𝔦𝔪𝔢𝔰F, we write S(F) for the set of primes of F above the primes in S:

S(F)=def{𝔭𝔓𝔯𝔦𝔪𝔢𝔰F𝔭FS}.

For FFF′′¯ subfields with F/F Galois and 𝔭𝔓𝔯𝔦𝔪𝔢𝔰F′′na, write D𝔭(F/F)Gal(F/F) for the decomposition group (i.e. the stabiliser) of 𝔭F𝔓𝔯𝔦𝔪𝔢𝔰Fna in Gal(F/F). (We sometimes write D𝔭=D𝔭(F/F), when no confusion arises.) Further, for FF¯ subfields with F/F Galois, set

Dec(F/F)=def{D𝔭(F/F)𝔭𝔓𝔯𝔦𝔪𝔢𝔰Fna}Sub(Gal(F/F)).

Thus, one has a canonical Gal(F/F)-equivariant surjective map 𝔓𝔯𝔦𝔪𝔢𝔰FnaDec(F/F), 𝔭D𝔭(F/F).

Given an algebraic extension K of and a nonarchimedean prime 𝔭 of K, we write κ(𝔭) for the residue field at 𝔭. Further, when K is a number field, we write K𝔭 for the completion of K at 𝔭, and, in general, we write K𝔭 for the union of K𝔭K for finite subextensions K/ of K/. For a subfield MK, we sometimes write M𝔭 instead of M𝔭M.

Given a number field K and a nonarchimedean prime 𝔭𝔓𝔯𝔦𝔪𝔢𝔰Kna above a prime p𝔓𝔯𝔦𝔪𝔢𝔰, K𝔭/p is a finite extension. We write d𝔭, e𝔭, f𝔭, and N(𝔭) for the local degree [K𝔭:p], the ramification index of K𝔭/p, the residual degree [κ(𝔭):𝔽p], and the norm |κ(𝔭)|, respectively, where 𝒪K is the ring of integers of K. (Thus, d𝔭=e𝔭f𝔭 and N(𝔭)=pf𝔭.)

Let K be a number field, and p a prime number which splits as (p)=i=1k𝔭ie𝔭i in K. Define the splitting type of p in K by (f𝔭1,,f𝔭k) ordered by f𝔭if𝔭i+1, which is a monotone non-decreasing finite sequence of positive integers. For each monotone non-decreasing finite sequence A of positive integers, write 𝒫K(A)𝔓𝔯𝔦𝔪𝔢𝔰na for the set of prime numbers with splitting type A in K. Two number fields K1,K2 are called arithmetically equivalent if 𝒫K1(A)=𝒫K2(A) for every such sequence A.

Let l be a prime number and set l~=defl (resp. l~=def4) for l2 (resp. l=2). Then the multiplicative group l× is canonically decomposed into the direct product

l×(1+l~l)×(l×)tor.

We denote the first projection l×1+l~l by αα¯ (αl×). More explicitly, we have α¯=α[αmodl~]-1, where we denote the Teichmüller lift (i.e. the unique group-theoretic section of l×(/l~)×) by β[β] (β(/l~)×).

Given a prime number l, a profinite group G, and a character χ:Gl×, we write χ¯:G1+l~l for the character defined by χ¯(g)=χ(g)¯ (gG).

Given a commutative ring R, an R-module M, and a subset S={m1,m2,} of M, we write SR=m1,m2,RM (or simply S if there is no risk of confusion) for the R-submodule of M generated by S. Given xM, we write

AnnR(x)=def{rRrx=0}

for the annihilator of x in R. We write

MR-tor=def{mMrm=0 for some non-zero-divisor rR}.

Given aR, we write (a)=aRR for the principal ideal of R generated by a. An R-submodule N of M is called R-cofinite if the quotient M/N is a finitely generated R-module.

1 The local theory

In this section we establish the local theory necessary to prove Theorems 1 and 2. We use the notations in the Introduction.

1.1 Structure of local Galois groups

Let K be a number field, 𝔭𝔓𝔯𝔦𝔪𝔢𝔰Kna a nonarchimedean prime above a prime p𝔓𝔯𝔦𝔪𝔢𝔰, 𝔭~ a prime of K¯ above 𝔭, and D𝔭~GK the decomposition group at 𝔭~. Thus, D𝔭~Gal(K¯𝔭~/K𝔭) is isomorphic to the absolute Galois group of K𝔭 (cf. [9, (8.1.5) Proposition]). We write

D𝔭~D𝔭~tameD𝔭~ur

for the maximal tame and unramified quotients of D𝔭~, respectively (cf. [9, discussion before (7.5.2) Proposition]), and set

I𝔭~=defKer(D𝔭~D𝔭~ur)andI𝔭~tame=defKer(D𝔭~tameD𝔭~ur).

For m0, let 𝔭~m be the image 𝔭~Km of 𝔭~ in Km and D𝔭~mGKm the decomposition group at 𝔭~m. Thus, we have a natural surjective homomorphism D𝔭~D𝔭~m which factors as

D𝔭~D𝔭~mD𝔭~m.

Proposition 1.

Let m0 be an integer. Then the following hold.

1. The surjective map D𝔭~mD𝔭~m is an isomorphism. In particular, the natural surjective maps Gal(K¯𝔭~/K𝔭)abD𝔭~1D𝔭~1 are isomorphisms.

2. If m1, then p is the unique prime number l such that logl|D𝔭~mab/tor/lD𝔭~mab/tor|2.

3. If m1, then d𝔭=logp|D𝔭~mab/tor/pD𝔭~mab/tor|-1.

4. If m1, then f𝔭=logp(1+|((D𝔭~mab)tor)(p)|), e𝔭=d𝔭/f𝔭, and N(𝔭)=pf𝔭.

5. If m1, then there is a factorisation D𝔭~D𝔭~mD𝔭~ur.

6. If m2, then there is a factorisation D𝔭~D𝔭~mD𝔭~tame, and Ker(D𝔭~mD𝔭~tame) is the maximal normal pro-p subgroup of D𝔭~m.

7. For each 2jm (resp. 0jm-1), the kernel of the projection D𝔭~mD𝔭~j is pro-p (resp. infinite).

8. If m2 and l is a prime number, then the inflation maps

H2(D𝔭~tame,𝔽l(1))H2(D𝔭~m,𝔽l(1))H2(D𝔭~,𝔽l(1))𝔽l

are isomorphisms for lp, and the inflation map

H2(D𝔭~m,𝔽l(1))H2(D𝔭~,𝔽l(1))𝔽l

is surjective for l=p.

9. If m2, then D𝔭~m is centre free.

10. If m2, then D𝔭~m is torsion free.

Proof.

(i) This follows, by induction on m0, from the fact that the natural map D𝔭~1GK1 is injective (cf. [3, III, 4.5 Theorem]) and applying this to finite extensions of K corresponding to various open subgroups of GKm-1.

(ii)–(iv) These assertions follow immediately from local class field theory.

(v) This follows from (i) and the fact that D𝔭~ur^ is abelian.

(vi) This follows from (i), the fact that D𝔭~tame is metabelian (cf. [9, (7.5.2) Proposition)], and Ker(D𝔭~D𝔭~tame) is the maximal normal pro-p subgroup of D𝔭~ (cf. [9, (7.5.7) Corollary (i)]).

(vii) Let 2jm. Then, by part (vi), there is a factorisation D𝔭~mD𝔭~jD𝔭~tame, and Ker(D𝔭~mD𝔭~j) is a subgroup of the pro-p group Ker(D𝔭~mD𝔭~tame). It follows that Ker(D𝔭~mD𝔭~j) is also pro-p. Next, let 0jm-1. Then Ker(D𝔭~mD𝔭~j) contains Ker(D𝔭~mD𝔭~m-1)=Ker(D𝔭~mD𝔭~m-1)=D𝔭~[m-1]ab, where the first equality follows from (i). So, it suffices to prove that D𝔭~[i]ab is infinite for i0. The case i=0 follows immediately from part (i) and local class field theory. Let D𝔭~I𝔭~I𝔭~tame (^(p)) be the inertia and the tame inertia groups, and I𝔭~wild=Ker(I𝔭~I𝔭~tame) the wild inertia group. By part (i) and local class field theory, Im(I𝔭~GK1) is a direct product of a finitely generated pro-p abelian group and a prime-to-p finite cyclic group. By part (v) (resp. (vi)), for i1 (resp. i2), D𝔭~[i]I𝔭~ (resp. D𝔭~[i]I𝔭~wild). It follows from these that the image of D𝔭~[1]=I𝔭~D𝔭~[1]I𝔭~tame is open in I𝔭~tame, hence D𝔭~[1]ab is infinite, and that for i2, D𝔭~[i]I𝔭~wild, hence D𝔭~[i] is a free pro-p group (cf. [5, Theorem 2 (ii)]), and D𝔭~[i]ab is a free pro-p abelian group. Thus, it suffices to prove D𝔭~[i]{1}. Suppose D𝔭~[i]={1}. Then D𝔭~=D𝔭~i is (i-step) solvable. This is absurd, as I𝔭~wildD𝔭~ is a free pro-p group of countable rank (cf. [5]), hence is not solvable.

(viii) If lp, this follows from the fact that Ker(D𝔭~D𝔭~tame), hence Ker(D𝔭~D𝔭~m) and Ker(D𝔭~mD𝔭~tame) are pro-p groups. If l=p, consider the commutative diagram

where the horizontal maps are cup products. By local Tate duality, the lower horizontal map gives a perfect pairing, hence is surjective (as H1(D𝔭~,𝔽p(1))=K𝔭×/(K𝔭×)p0). Thus, it suffices to show that the natural inflation maps H1(D𝔭~m,𝔽p(i))H1(D𝔭~,𝔽p(i)), i=0,1 are surjective. Let χcycl(mod p):D𝔭~𝔽p× be the mod p cyclotomic character (which factors through D𝔭~m) and

Δ𝔭~=defIm(χcycl(mod p)).

Write N𝔭~m=defKer(D𝔭~mΔ𝔭~), and N𝔭~=defKer(D𝔭~Δ𝔭~). Then we have the commutative diagram

where the horizontal and vertical maps are inflation and restriction maps, respectively. Here, the vertical maps are isomorphisms since |Δ𝔭~| is prime to p (as it divides p-1). The lower horizontal map is also an isomorphism since N𝔭~abN𝔭~mab (use (i) and m2), and both N𝔭~ and N𝔭~m act trivially on 𝔽p(i). Thus, the upper horizontal map is an isomorphism, as desired.

(ix) We prove this by induction on m2. If m=2, this follows from (i) and [6, Theorem 9.3]. If m>2, then it follows from the induction hypothesis for m-1 that

C(D𝔭~m)Ker(D𝔭~mD𝔭~m-1)Ker(D𝔭~mD𝔭~m-2).

Now, applying the m=2 case to the maximal metabelian quotients of open subgroups of D𝔭~m obtained as the inverse image of various open subgroups of D𝔭~m-2, one sees C(D𝔭~m)={1}.

(x) For each i1, (D𝔭~:D𝔭~[i]) is divisible by l for every prime number l (cf. (v)), hence cd(D𝔭~[i])1, which implies that

D𝔭~[i+1,i]=D𝔭~[i]ab

is torsion free. (Observe D𝔭~[i]ab=l𝔓𝔯𝔦𝔪𝔢𝔰na(D𝔭~[i](l))ab and that for each l𝔓𝔯𝔦𝔪𝔢𝔰na, cd(D𝔭~[i])1 implies cd(D𝔭~[i](l))1, hence D𝔭~[i](l) is a free pro-l group.) Similarly, as |D𝔭~ur| is divisible by l for every prime number l, one has cd(I𝔭~)1. This implies that I𝔭~ab (which is a subquotient of D𝔭~2, as D𝔭~ur is abelian) is torsion free. Let H be any finite subgroup of D𝔭~m. As D𝔭~ur^ is torsion free, H is contained in Im(I𝔭~D𝔭~m)=I𝔭~/(D𝔭~[m]). As I𝔭~ab is torsion free and m2, H is contained in Im(I𝔭~[1]D𝔭~m)Im(D𝔭~[1]D𝔭~m)=D𝔭~[m,1]. As D𝔭~[i+1,i] is torsion free for i=1,,m-1, H must be trivial, as desired. ∎

1.2 Separatedness in GKm

Let K be a number field, let 𝔭,𝔭𝔓𝔯𝔦𝔪𝔢𝔰K¯na, and let D𝔭,D𝔭GK the decomposition groups at 𝔭,𝔭, respectively. Then it is well known (cf. [9, (12.1.3) Corollary]) that the following separatedness property holds:

D𝔭D𝔭{1}𝔭=𝔭.

This does not hold as it is, if K¯ and GK are replaced by Km and GKm, respectively (cf. Propositions 3 and 13). However, we show certain weaker separatedness properties hold in the latter situation.

Lemma 2.

Let G be a profinite group, A an abelian normal closed subgroup of G, and F an infinite closed subgroup of G. Then, for each open subgroup H of G containing A, there exists an open subgroup H of H containing A, such that Im(FH(H)ab){1}.

Proof.

As F is infinite and H is open in G, FH is nontrivial.

Case 1: FHA, i.e. Im(FHG/A){1}. In this case, there exists a normal open subgroup N of G with ANG such that Φ=defIm(FHG/N){1}. Take a nontrivial abelian (e.g. cyclic) subgroup Φ of Φ, and let H be the inverse image of Φ under HH/N. Then H is an open subgroup of H containing NA, and

Im(FHG/N)=Φ.

Now, as Φ is abelian, the natural map HΦ factors as H(H)abΦ. Since

Im(FHG/N)=Φ{1},

one has Im(FH(H)ab){1}, a fortiori.

Case 2: FHA. In this case, one has

A=AHopenHH=limAHopenHH,

hence

A=Aab=limAHopenH(H)ab.

Since FHA is nontrivial, this shows that there exists an open subgroup H of H containing A such that Im(FH(H)ab){1}. This completes the proof, as

FH(FH)H=FH.

Proposition 3.

Let m1 be an integer, let K be a number field, let p,pPrimesKmna, let Dp,DpGKm the decomposition groups at p,p, respectively, and let p¯,p¯ be the images of p,p in PrimesKm-1na, respectively. Then the following are equivalent:

1. D𝔭D𝔭{1}.

2. D𝔭D𝔭GK[m,m-1]{1}.

3. D𝔭GK[m,m-1] and D𝔭GK[m,m-1] are commensurable.

4. D𝔭GK[m,m-1]=D𝔭GK[m,m-1].

5. 𝔭¯=𝔭¯.

Proof.

The implications (ii)  (i′′′)  (i′′) and (i)  (i) are immediate. The implication (i′′)  (i) follows from the fact that D𝔭GK[m,m-1] and D𝔭GK[m,m-1] are infinite as follows from Proposition 1 (vii). The implication (i)  (ii) for m=1 follows from [3, III, 4.16.7 Corollary]. Thus, we prove the implication (i)  (ii) for m2. For this, assume that (i) holds, and set F=defD𝔭D𝔭({1}). As D𝔭 is torsion free by Proposition 1 (x), F is infinite. Let M/K be any finite subextension of Km-1/K and set H=defGal(Km/M)GKm. Then H is an open subgroup of GKm containing A=defGK[m,m-1]GKm. By Lemma 2, there exists an open subgroup H of H containing A such that Im(FH(H)ab){1}. Set

M=defKmHKmGK[m,m-1]=Km-1andM1=defKmH[1]Km.

As M is contained in Km-1, M1 coincides with the maximal abelian extension (M)ab of M. Write 𝔭M1,𝔭M1 for the images of 𝔭,𝔭 in 𝔓𝔯𝔦𝔪𝔢𝔰M1na, respectively, 𝔭¯M,𝔭¯M for the images of 𝔭¯,𝔭¯ in 𝔓𝔯𝔦𝔪𝔢𝔰Mna, respectively, 𝔭¯M,𝔭¯M for the images of 𝔭¯,𝔭¯ in 𝔓𝔯𝔦𝔪𝔢𝔰Mna, respectively, and D,D(H)ab=Gal(M1/M) for the decomposition groups at 𝔭M1,𝔭M1, respectively. Now, since

DDIm(D𝔭D𝔭H(H)ab)=Im(FH(H)ab){1},

one obtains 𝔭¯M=𝔭¯M by applying the implication (i)  (ii) for m=1 (which we have already established) to the number field M. Thus, one has 𝔭¯M=𝔭¯M, a fortiori. As M/K is an arbitrary finite subextension of Km-1/K, this shows that 𝔭¯=𝔭¯, as desired. ∎

Lemma 4.

Let G be a profinite group and H a closed subgroup of G. Let l be a prime number. Consider the following conditions (i)–(v):

1. H is l-open in G (cf. Notations).

2. For every l-Sylow subgroup LH of H, there exists an l-Sylow subgroup L of G containing LH such that LH is open in L.

3. For every l-Sylow subgroup LH of H and every l-Sylow subgroup L of G containing LH, LH is open in L.

4. l(G:H) i.e. there exists an integer N>0 such that for any surjective homomorphism π:GG¯ with G¯ finite, lN(G¯:H¯), where H¯=defπ(H).

5. For any surjective homomorphism π:GG¯ with G¯ almost pro-l (i.e. admitting an open pro-l subgroup), H¯=defπ(H) is open in G¯.

Then one has (i)  (ii)  (iii)  (iv)  (v).

Proof.

Omitted. ∎

The main result in this subsection is the following.

Proposition 5.

Let K be a number field, and K~/K a (an infinite) Galois extension such that QabK~. Then the natural surjective map: PrimesK~abnaDec(K~ab/K) is bijective.

Corollary 6.

Let K be a number field and m2 an integer. Then the natural surjective map PrimesKmnaDec(Km/K) is bijective.

Proof of Corollary 6.

Apply Proposition 5 to K~=Km-1. ∎

Corollary 7.

With the notations and assumptions in Proposition 5, the centraliser of Gal(K~ab/K) in Aut(K~ab) is trivial, and Gal(K~ab/K) is centre free. In particular, for m2, the centraliser of GKm in Aut(Km) is trivial, and GKm is centre free.

Proof of Corollary 7.

By definition, the natural action (by conjugacy) on Dec(K~ab/K) of the centraliser of Gal(K~ab/K) in Aut(K~ab) is trivial, hence, by Proposition 5, that on 𝔓𝔯𝔦𝔪𝔢𝔰K~abna is trivial. The first assertion follows from this, together with Lemma 8 below (applied to E=F=K~). The second assertion follows from the first. The third and the fourth assertions follow from the first and the second, applied to K~=Km-1. ∎

Lemma 8.

Let E,F be algebraic extensions of Q. Then the natural map

Isom(fields)(E,F)Isom(sets)(𝔓𝔯𝔦𝔪𝔢𝔰Ena,𝔓𝔯𝔦𝔪𝔢𝔰Fna)

is injective.

Proof.

Let σ1,σ2Isom(fields)(E,F) and assume that they induce the same bijection 𝔓𝔯𝔦𝔪𝔢𝔰Ena𝔓𝔯𝔦𝔪𝔢𝔰Fna. Set σ=defσ1σ2-1Aut(F) and F0=defFσ. Thus, F/F0 is a (possibly infinite) Galois extension with Galois group σ¯. Let F0/F0 be any finite subextension of F/F0. Then F0/F0 is a finite cyclic extension with Galois group σ0, where σ0=defσ|F0. Further, there exists a finite subextension F00/ of F0/ (which may depend on F0/F0) and a finite cyclic extension F00/F00 contained in F0 such that F00F00F0F0. Note that Gal(F00/F00)=σ00, where σ00=σ0|F00.

By assumption, σ induces the identity on 𝔓𝔯𝔦𝔪𝔢𝔰Fna, hence σ0 and σ00 induces the identity on 𝔓𝔯𝔦𝔪𝔢𝔰F0na and 𝔓𝔯𝔦𝔪𝔢𝔰F00na, respectively. Now, by Chebotarev’s density theorem, there exists a prime 𝔭𝔓𝔯𝔦𝔪𝔢𝔰F00na which splits completely in F00/F00. Then Gal(F00/F00)=σ00 acts regularly (i.e. transitively and freely) on the set of primes in 𝔓𝔯𝔦𝔪𝔢𝔰F00na above 𝔭. Since this action is also trivial, one must have F00=F00, hence F0=F0. Because F0/F0 is arbitrary, we conclude F=F0, i.e. σ=1, as desired. ∎

Proposition 5 follows immediately from the implication (ii)  (i) in the following Proposition 9.

Proposition 9.

With the notations and assumptions in Proposition 5, assume that p,pPrimesK~abna. Write Dp=defDp(K~ab/K) and Dp=defDp(K~ab/K). Consider the following conditions:

1. 𝔭=𝔭.

2. D𝔭=D𝔭.

3. D𝔭,D𝔭 are commensurable.

4. For every prime number l, D𝔭,D𝔭 are l-commensurable.

5. For every prime number l but one l0, D𝔭,D𝔭 are l-commensurable.

6. For some prime number l, D𝔭,D𝔭 are l-commensurable.

7. 𝔭K~=𝔭K~.

Then one has (i)  (ii)  (iii)  (iv)  (v)  (vi)  (vii).

Proof.

Here, the implications (i)  (ii)  (iii)  (iv)  (v)  (vi) follow immediately, while the implication (vi)  (vii) follows from [3, III, (4.6.17) Corollary] (applied to various finite subextensions of K~/K), as in the proof of Proposition 3, (i)  (ii). So, we may concentrate on proving the implication (iv)  (i). We start with some lemmas.

Lemma 10.

Let F be a number field, let F/F be a finite abelian extension, and let q,qPrimesFna, with qq. Then there exists a subextension F′′/F of F/F such that F′′/F is cyclic of prime power order; qF′′qF′′; and qF, qF split completely in F′′/F.

Proof of Lemma 10.

If 𝔮F𝔮F, we may take F′′=F. So, we may assume that 𝔮F=𝔮F, which implies that there exists σG=defGal(F/F) such that 𝔮=σ𝔮. As F/F is abelian, one has D=defD𝔮(F/F)=D𝔮(F/F). As 𝔮𝔮, one has σD. Let σ¯ be the image of σ in the quotient group G/D. Write G/DC1Cr, where Ci (i=1,,r) is a cyclic subgroup of prime power order. As σ¯1, there exists at least one i such that the image of σ¯ under the projection G/DCi is nontrivial. Now the subextension F′′/F of F/F corresponding to the quotient GG/DCi satisfies the conditions. ∎

Lemma 11.

Let F be a number field, and qPrimesFna. For each integer m>0, write ψF,q,m for the natural map F×/(F×)mFq×/(Fq×)m induced by the natural inclusion F×Fq×. Then, for each pair of positive integers m,n, the natural map

Ker(ψF,𝔮,mn)Ker(ψF,𝔮,m)

induced by the natural projection F×/(F×)mnF×/(F×)m is surjective.

Proof of Lemma 11.

Consider the commutative diagram

in which the two rows are exact. (Here, the first horizontal arrows are induced by the m-th power maps and the second horizontal arrows are the natural projections.) Observe that the vertical arrows are surjections. Indeed, for each integer k>0, (F𝔮×)k is open in F𝔮×, and the image of the natural inclusion F×F𝔮× is dense. Now, the assertion follows from diagram-chasing. ∎

Proof of Proposition 9, (iv) ⇒ (i).

Assume that (iv) holds. (Then (vii) holds, a fortiori, i.e. 𝔭K~=𝔭K~.) Suppose that (i) does not hold, i.e. 𝔭𝔭. Then there exists a finite subextension L/K of K~ab/K such that 𝔭L𝔭L. Set M=defLK~. Then M/K is a finite subextension of K~/K and L/M is a finite extension. As MK~, one has 𝔭M=𝔭M. Note that we may replace L (and M, correspondingly) by any finite extension of L contained in K~ab.

Step 1. Reduction: We may assume that L/M is abelian. Indeed, if we replace L by the Galois closure L1 of L/M (and M by M1=defL1K~), this holds.

Step 2. Reduction: We may assume that L/M is cyclic of order lr for some prime number l and some integer r0 and that 𝔭M splits completely in L/M. Indeed, this follows from Lemma 10, applied to L/M.

Step 3. Reduction: We may assume that the conditions of Step 2 and the condition that ζlrM hold. Indeed, if we replace L by L(ζlr) (and M by L(ζlr)K~), this holds with r replaced by r below. First, one has M(ζlr)L(ζlr)MabL(ζlr)K~. Next, consider the commutative diagram of fields:

LL(ζlr)||L(ζlr)K~|MM(ζlr).

As L/M is cyclic of order lr and 𝔭M splits completely in L/M, we see that L(ζlr)/L(ζlr)K~ is cyclic of order lr with rr and 𝔭L(ζlr)K~ splits completely in L(ζlr)/L(ζlr)K~. (Observe Gal(L(ζlr)/(L(ζlr)K~))Gal(L(ζlr)/M(ζlr))Gal(L/M).) As

ζlrM(ζlr)L(ζlr)K~,

we are done.

Step 3 is the final reduction step and we will not replace L (or M) any more.

Step 4: r>0. Indeed, otherwise, L=M, which contradicts 𝔭L𝔭L and 𝔭M=𝔭M.

Step 5: L/M is linearly disjoint from M(ζl)/M. Indeed, one has

MLM(ζl)LMabLK~=M.

Step 6: The l-adic cyclotomic character and l-commensurability. Let

χcycl(l):GKl×

denote the cyclotomic character. As K(ζl)KabK~, it follows that χcycl(l) factors through Gal(K~/K), hence, a fortiori, through Gal(K~ab/K). By abuse, write χcycl(l) again for the l-adic character of Gal(K~ab/K) induced by χcycl(l). Since the number of l-power roots of unity in each finite extension of M𝔭 is finite (as M𝔭 is a finite extension of p, where p is the characteristic of κ(𝔭)), χcycl(l)(D𝔭(K~ab/M))l× is open. Further, as D𝔭 and D𝔭 are l-commensurable, it follows that the subgroup χcycl(l)(D𝔭(K~ab/M)D𝔭(K~ab/M))l× is open. So, we may take

σ~D𝔭(K~ab/M)D𝔭(K~ab/M)

such that χcycl(l)(σ~)1, and χcycl(l)(σ~)=1+u0lr0 with r00 and u0l×. (As ζlrM, one has r0r>0.)

Step 7: Kummer theory. As L/M is a cyclic extension of order lr and ζlrM, there exists fM×(M×)l such that L=M(f1/lr). As 𝔭M splits completely in L/M, one has f(M𝔭×)lr. Then the image fmod(M×)lr of f in M×/(M×)lr lies in the kernel of the natural map

ψM,𝔭M,lr:M×/(M×)lrM𝔭×/(M𝔭×)lr.

By Lemma 11, there exists an element γ of the kernel of

ψM,𝔭M,lr+r0:M×/(M×)lr+r0M𝔭×/(M𝔭×)lr+r0

which maps to fmod(M×)lr under the projection M×/(M×)lr+r0M×/(M×)lr. Take any gM× whose image in M×/(M×)lr+r0 is γ.

As fmod(M×)lr=gmod(M×)lrM×/(M×)lr, one has

L=M(f1/lr)=M(g1/lr).

We set M=defM(ζlr+r0), L=defLM=M(f1/lr)=M(g1/lr), and L′′=defM(g1/lr+r0). Then MMabK~. By Step 5, L/M is cyclic of order lr, hence g(M)×((M)×)l. This, together with the fact that ζlr+r0M, implies L′′/M is cyclic of order lr+r0. Further, as gM×, the extension L′′/M is Galois. By the choice of g, the image of g in M𝔭×/(M𝔭×)lr+r0 is trivial, hence the image of g in (M𝔭)×/((M𝔭)×)lr+r0 is trivial, a fortiori. Thus, 𝔭M splits completely in L′′/M.

Step 8: End of proof. Consider the following commutative diagram in which the two rows are exact:

Here, as 𝔭M splits completely in L′′/M, we have D𝔭(L′′/M)=1. Hence the projection Gal(L′′/M)Gal(M/M) induces an isomorphism D𝔭(L′′/M)D𝔭(M/M). Also, one has a natural injection Gal(M/M)(/lr+r0)× induced by the modulo lr+r0 cyclotomic character, and an isomorphism Gal(L′′/M)/lr+r0(1) as Gal(M/M)-modules. (Indeed, observe the commutative diagram

in which the vertical arrows are induced by the natural inclusion MM, the horizontal isomorphisms arise from Kummer theory, and the equality follows from the fact that ζlr+r0M. Here, the image of gmod(M×)lr+r0M×/(M×)lr+r0 in H1(GM,/lr+r0(1))Gal(M/M) lies in IsomGal(M/M)(Gal(L′′/M),/lr+r0(1)), hence gives a Gal(M/M)-equivariant isomorphism Gal(L′′/M)/lr+r0(1).)

Define

σD𝔭(L′′/M)D𝔭(L′′/M)

to be the image of σ~D𝔭(K~ab/M)D𝔭(K~ab/M) under the natural surjective homomorphism Gal(K~ab/M)Gal(L′′/M). Note that the image of σ in

Gal(M/M)(/lr+r0)×=(l/lr+r0l)×

is 1+u0lr0modlr+r0, which acts on Gal(L′′/M)/lr+r0 by multiplication.

Next, one has 𝔭L′′𝔭L′′ (as 𝔭L𝔭L) and 𝔭M=𝔭M (as MK~). Accordingly, there exists τGal(L′′/M){1} such that 𝔭L′′=τ𝔭L′′, hence

D𝔭(L′′/M)=τD𝔭(L′′/M)τ-1.

In particular, as σD𝔭(L′′/M),

τ-1σττ-1D𝔭(L′′/M)τ=D𝔭(L′′/M).

Thus, σ,τ-1στD𝔭(L′′/M).

As the image of τGal(L′′/M) in Gal(M/M) is trivial, it follows that the images of σ,τ-1στGal(L′′/M) in Gal(M/M) coincide with each other. But as the projection Gal(L′′/M)Gal(M/M) induces an isomorphism D𝔭(L′′/M)D𝔭(M/M), we conclude σ=τ-1στ, hence

τGal(L′′/M)σ(/lr+r0)1+u0lr0=lr/lr+r0=lr(/lr+r0),

i.e. τGal(L′′/M)lr=Gal(L′′/L). This implies 𝔭L=τ𝔭L=𝔭L, a contradiction. ∎

Thus Proposition 9 is proved. ∎

In Proposition 9, the implication (v)  (i) fails in general. (For example, it fails in the case K~=Km-1 treated in Corollary 6.) More precisely, we show Proposition 13 below. Let P0 be a subset of 𝔓𝔯𝔦𝔪𝔢𝔰na.

Definition 12.

(i) We say that a profinite group is P0-perfect if it admits no nontrivial pro-l abelian quotient for any lP0.

(ii) We say that a field F¯ is P0-perfect if Gal(¯/F) is P0-perfect.

Proposition 13.

Assume P0PrimesQna. Then, with the assumptions in Proposition 5, the following are equivalent:

1. For each pair 𝔭,𝔭𝔓𝔯𝔦𝔪𝔢𝔰K~abna such that D𝔭 and D𝔭 are l-commensurable for every l𝔓𝔯𝔦𝔪𝔢𝔰naP0, one has 𝔭=𝔭.

2. K~ is P0-perfect.

Proof.

Note that the implication (ii)  (i) just follows from the proof of Proposition 9, (iv)  (i). (The assumption P0𝔓𝔯𝔦𝔪𝔢𝔰na is used to ensure (vii) there, via (vi) there.) More precisely, the only new input is the observation that, under the assumption that (ii) holds, l in Step 2 there automatically belongs to 𝔓𝔯𝔦𝔪𝔢𝔰naP0. Indeed, otherwise, i.e. if lP0, one must have LK~, since L/M is cyclic of l-power order, MK~, and K~ is P0-perfect.

To prove the implication (i)  (ii), suppose that K~ is not P0-perfect. Then there exists l0P0 such that K~ admits a finite cyclic extension L~ of degree l0, which then descends to a finite cyclic extension L/M of degree l0 over some field M with KMK~ and [M:K]<. (Thus, in particular, L~=LK~.) By Chebotarev’s density theorem, 𝔭M splits completely in L/M for some 𝔭PK~ab. We fix such 𝔭.

Next, as D𝔭(K~ab/M) is a quotient of GM𝔭, it is prosolvable. So, there exists an l0-Hall subgroup D of D𝔭(K~ab/M), that is, D is pro-prime-to-l0 and (D𝔭(K~ab/M):D) is a (possibly infinite) power of l0.

Now, consider the exact sequence of profinite abelian groups

1Gal(K~ab/L~)Gal(K~ab/K~)Gal(L~/K~)1,

which induces an exact sequence of pro-l0 abelian groups

1Gal(K~ab/L~)l0Gal(K~ab/K~)l0Gal(L~/K~)1,

where ()l0 refers to the l0-Sylow subgroups. As Gal(L~/K~)=Gal(LK~/K~)Gal(L/M), the Galois group Gal(K~ab/M) acts (by conjugation) trivially on Gal(L~/K~), hence, a fortiori, so does DD𝔭(K~ab/M)Gal(K~ab/M). But as D is a pro-prime-to-l0 group, the sequence obtained by taking the D-fixed parts

1(Gal(K~ab/L~)l0)D(Gal(K~ab/K~)l0)DGal(L~/K~)D=Gal(L~/K~)1

remains exact. Thus, a generator of the Galois group Gal(L~/K~)/l0 lifts to an element τ of (Gal(K~ab/K~)l0)DGal(K~ab/K~)D. As the image of τ in Gal(L/M) is nontrivial and 𝔭M splits completely in L/M, one has τD𝔭(K~ab/M), or, equivalently, 𝔭=defτ𝔭𝔭. On the other hand, τDτ-1=D as D fixes (i.e. commutes with) τ. Note that D (resp. τDτ-1=D) is an l0-Hall subgroup of D𝔭(K~ab/M) (resp. τD𝔭(K~ab/M)τ-1=D𝔭(K~ab/M)). Namely, D is an l0-Hall subgroup of both D𝔭(K~ab/M) and D𝔭(K~ab/M), which implies that D𝔭(K~ab/M) and D𝔭(K~ab/M) are l-commensurable for every l𝔓𝔯𝔦𝔪𝔢𝔰na{l0}, hence, in particular, for every l𝔓𝔯𝔦𝔪𝔢𝔰naP0. This contradicts (i), as 𝔭𝔭. ∎

1.3 Correspondence of decomposition groups

Finally, we establish local theory in our context. More precisely, starting from an isomorphism of the maximal (m+2)-step solvable Galois groups of number fields, we establish a one-to-one correspondence between the sets of their nonarchimedean primes and a one-to-one correspondence between the corresponding decomposition groups (cf. Corollary 27). This is achieved via a purely group-theoretic characterisation of decomposition groups (cf. Theorem 25).

As in the proof of the Neukirch–Uchida theorem, our proof is based on the following local-global principle.

Proposition 14.

Let K be a number field, and let l be a prime number. For a prime pPrimesK, let DpGK be a decomposition group at p (Dp is only defined up to conjugation). Then the following hold:

1. If 𝔭𝔓𝔯𝔦𝔪𝔢𝔰Kna, then H2(D𝔭,𝔽l(1))𝔽l.

2. If l is odd or K is totally imaginary, then there exists a natural injective homomorphism

ψK:H2(GK,𝔽l(1))𝔭H2(D𝔭,𝔽l(1)),

where the direct sum is over all nonarchimedean primes 𝔭𝔓𝔯𝔦𝔪𝔢𝔰Kna.

Proof.

Assertions (i) and (ii) are well known (cf. [9, (8.1.5) Proposition, (7.1.8) Theorem (ii), (9.1.10) Corollary], and the fact that H2(D𝔭,𝔽l(1))=0 if 𝔭 is archimedean and either l2 or K is totally imaginary). ∎

Corollary 15.

Let L be an algebraic extension of Q, and l a prime number. For a prime pPrimesL, let DpGL be a decomposition group at p (Dp is only defined up to conjugation). Then the following hold:

1. If 𝔭𝔓𝔯𝔦𝔪𝔢𝔰Lna, then H2(D𝔭,𝔽l(1))𝔽lϵ with ϵ1. Further, ϵ=1 if and only if D𝔭 is l-open in Dp, where p𝔓𝔯𝔦𝔪𝔢𝔰 is the characteristic of κ(𝔭).

2. If l is odd or L is totally imaginary, then there exists a natural injective homomorphism

ψL:H2(GL,𝔽l(1))𝔭H2(D𝔭,𝔽l(1)),

where the product is over all nonarchimedean primes 𝔭𝔓𝔯𝔦𝔪𝔢𝔰Lna.

Proof.

The first half of assertion (i) and assertion (ii) are reduced to assertion (i) and assertion (ii) of Proposition 14, respectively, by taking the inductive limits. The second half of assertion (i) also follows from assertion (i) of Proposition 14, and the fact that for open subgroups DDDp(Gp), the restriction map from H2(D,𝔽l(1))=𝔽l to H2(D,𝔽l(1))=𝔽l is the (D:D)-multiplication. ∎

The following lemma will be of important use later.

Lemma 16.

Let G be a profinite group, m0 an integer and N a finite discrete Gm-module. For a closed subgroup FGm, write F~ (resp. F~1) for the inverse image of F in G (resp. in Gm+1). Then the natural map

2(F,N)=defIm(H2(F,N)infH2(F~1,N))infH2(F~,N)

is injective.

Proof.

We have the following commutative diagram:

where the horizontal sequences are the exact sequences arising from the Hochschild–Serre spectral sequences, the vertical maps are inflation maps,

G[m]=Ker(GGm)=Ker(F~F)

and

G[m+1,m]=Ker(Gm+1Gm)=Ker(F~1F).

The left vertical map is an isomorphism since G[m]abG[m+1,m] and both G[m] and G[m+1,m] act trivially on N. The middle vertical map is an isomorphism since it is the identity. Now, our assertion follows by an easy diagram chasing. ∎

The following result is the main step towards establishing the desired local theory in our context.

Proposition 17.

Let K be a number field, and l a prime number. Let FGKm be a closed subgroup. (We use the notations in Lemma 16 for G=GK and F.) Set L=def(Km)F, and assume that either l is odd or L is totally imaginary. For each nonarchimedean prime p~PrimesLna, write F~p~F~=GL for a decomposition group at p~ (defined up to conjugation in F~). Then there exists a natural injective homomorphism

2(F,𝔽l(1))𝔭~H2(F~𝔭~,𝔽l(1)),

where the product is over all nonarchimedean primes p~PrimesLna.

Proof.

We have natural maps

2(F,𝔽l(1))H2(F~,𝔽l(1))𝔭~H2(F~𝔭~,𝔽l(1)),

where the first map is an inflation map and the second map is a product of restriction maps. The first map is injective by Lemma 16 and the second map is injective by Corollary 15 (ii). Thus, the composite is also injective, as desired. ∎

Definition 18.

Let l be a prime number. Given a profinite group F, we say that F is an l-decomposition like group if there exists an exact sequence 1F1FF21, where F1,F2 are free pro-l of rank 1 (i.e. isomorphic to l).

Lemma 19.

Let F be an l-decomposition like group and D a closed subgroup of F. Consider the following conditions:

1. D=F.

2. The restriction map H2(F,𝔽l)H2(D,𝔽l) is an isomorphism.

3. The restriction map H2(F,𝔽l)H2(D,𝔽l) is nontrivial.

4. D is open in F.

5. D is an l-decomposition like group.

6. H2(D,𝔽l)𝔽l.

7. H2(D,𝔽l)0.

8. cdl(D)=2.

Then one has (i)  (ii)  (iii)  (iv)  (v)  (vi)  (vii)  (viii).

Proof.

The three implications (i)  (ii), (i)  (iv), and (vi)  (vii) are trivial. We prove (v)  (vi). As D is an l-decomposition like group, there exists an exact sequence

1D1DD21

with D1,D2l. By the Hochschild–Serre spectral sequence and the fact that cdl(l)=1 and H1(l,𝔽l)𝔽l, one has H2(D,𝔽l)H1(D2,H1(D1,𝔽l))H1(D2,𝔽l)𝔽l, as desired. (For the second isomorphism, observe that the isomorphism H1(D1,𝔽l)𝔽l is automatically D2-equivariant, since any homomorphism from the pro-l group D2 to 𝔽l× is trivial.) In particular, by applying (v)  (vi) to D=F, one has H2(F,𝔽l)𝔽l. Thus, the implication (ii)  (iii) follows.

Next, as F is an l-decomposition like group, there exists an exact sequence

1F1FF21

with F1,F2l. Set D1=defDF1 and D2=defIm(DF2). Then, if the restriction map from H2(F,𝔽l)(H1(F2,H1(F1,𝔽l))) to H2(D,𝔽l)(H1(D2,H1(D1,𝔽l))) is nontrivial, one must have D1=F1 and D2=F2, hence D=F. This shows (iii)  (i).

Next, if D is open in F, then D1 and D2 must be open in F1 and F2, respectively. As any open subgroup of l is isomorphic to l, this shows (iv)  (v).

Further, as H2(D,𝔽l)H1(D2,H1(D1,𝔽l)), (vii) implies D11 and D21. As any nontrivial closed subgroup of l is open, this shows (vii)  (iv).

Finally, as D is an extension of D2F2l by D1F1l, one has cdl(D)2. Thus, (vii)  (viii) follows (cf. [9, (3.3.2) Proposition]). This completes the proof. ∎

Definition 20.

Let m2 be an integer, FGKm a closed subgroup, and l a prime number. Then we say that F satisfies condition (l) if the following two conditions hold.

1. F is an l-decomposition like group (cf. Definition 18).

2. With the notations in Lemma 16, 2(F,𝔽l)0.

Remark 21.

We use the notations in Definition 20.

1. Assume that F satisfies condition (l) (a). Then the F-module 𝔽l(1) is isomorphic to the trivial module 𝔽l, as any homomorphism from the pro-l group F to 𝔽l× is trivial. This, together with Lemma 19, (i)  (vi), shows that condition (l) (b) is equivalent to saying that 2(F,𝔽l(1))𝔽l.

2. Condition (l) is group-theoretic in the following sense: One can detect purely group-theoretically whether or not a closed subgroup FGKm satisfies (l)if we start from (the isomorphy type of) GKm+1.

Proposition 22.

Let m2 be an integer, FGKm a closed subgroup, and l a prime number. Then F satisfies condition (l) if and only if F is an open subgroup of an l-Sylow subgroup of the decomposition group DpGKm at some pPrimesKmna with residue characteristic l. Further, then the image p¯PrimesKm-1na of p is uniquely determined by F.

Proof.

First, we prove the “only if” part of the first assertion. So, assume that F is an open subgroup of an l-Sylow subgroup of the decomposition group D𝔭GKm at some 𝔭𝔓𝔯𝔦𝔪𝔢𝔰Kmna with residue characteristic pl. By Proposition 1 (vi), F is isomorphically mapped onto an open subgroup F¯ of an l-Sylow subgroup of D𝔭~tame (where 𝔭~ is any element of 𝔓𝔯𝔦𝔪𝔢𝔰K¯ above 𝔭) by the surjection D𝔭D𝔭~tame. Note that there exists an exact sequence

1I𝔭~tameD𝔭~tameD𝔭~ur1

with D𝔭~ur^ and I𝔭~tame^(p). This implies that F(F¯) satisfies (l) (a). Again by Proposition 1 (vi), there exists a closed subgroup F of D𝔭~ which is isomorphically mapped onto F by the surjection D𝔭~D𝔭. The isomorphism FF factors as FF~F, where FF~ is the natural inclusion and F~F is the natural surjection induced by the surjection GKGKm. Accordingly, the isomorphism H2(F,𝔽l)H2(F,𝔽l) induced by the isomorphism FF is the composite of the inflation map H2(F,𝔽l)H2(F~,𝔽l) and the restriction map H2(F~,𝔽l)H2(F,𝔽l). In particular, H2(F,𝔽l)H2(F~,𝔽l) is injective, which implies H2(F,𝔽l)2(F,𝔽l). Now, by the fact that F satisfies condition (l) (a), together with Lemma 19, (i)  (vi), one has 2(F,𝔽l)(H2(F,𝔽l))𝔽l0. Thus, F satisfies (l) (b).

Next, we prove the “if” part of the first assertion. So, assume that F satisfies condition (l). Let L=def(Km)F. We claim that L is totally imaginary. Indeed, otherwise, one has an embedding L, which extends to an embedding Km. To these embeddings, a homomorphism Gal(/)GKm is associated. As Kmm1(-1), this homomorphism is injective. This is absurd, since F is torsion free as it satisfies (l) (a). Together with this, Proposition 17 and Remark 21 (i) imply that we have an injective homomorphism

2(F,𝔽l)𝔭~H2(F~𝔭~,𝔽l),

where the product is over all nonarchimedean primes 𝔭~𝔓𝔯𝔦𝔪𝔢𝔰Lna. This map is nontrivial since F satisfies condition (l) (b). Thus, there exists 𝔭~𝔓𝔯𝔦𝔪𝔢𝔰L such that the map 2(F,𝔽l)H2(F~𝔭~,𝔽l) is nontrivial. In particular, H2(F,𝔽l)H2(F𝔭~,𝔽l) and the group H2(F~𝔭~,𝔽l(1)) (H2(F~𝔭~,𝔽l)) are nontrivial, where F𝔭~=defIm(F~𝔭~GKm) is a decomposition subgroup of F=Gal(Km/L) at 𝔭~. The former nontriviality, together with Lemma 19, (iii)  (i), implies that F=F𝔭~. The latter nontriviality, together with Corollary 15 (i), implies that F~𝔭~ is l-open in a decomposition subgroup of G at p, where p is the residue characteristic of 𝔭~. In particular, F=F𝔭~ is l-open in a decomposition subgroup D𝔭 of GKm at 𝔭 (where 𝔭𝔓𝔯𝔦𝔪𝔢𝔰Kna stands for the image of 𝔭~), as desired. Next, take a finite abelian extension K of K𝔭 with [K:p]>1, which corresponds to an open subgroup H of D𝔭 containing D𝔭[1] (cf. Proposition 1 (i)). As FH is l-open in H, the natural map

(FH)abllHab^l

must be surjective. Here, by Lemma 19, (iv)  (v), FH is an l-decomposition like group, hence (topologically) generated by two elements and diml((FH)abll)2. On the other hand, by local class field theory, together with the fact that m2, dim(Hab^l)=1 (resp. [K:p]+1>2) if lp (resp. l=p). Therefore, one must have lp, as desired. Finally, the second assertion follows immediately from Proposition 3. This completes the proof. ∎

Let m2 be an integer. For a prime number l, write

𝒟~m,l=def{FSub(GKm)F satisfies (l)}.

On the set 𝒟~m,l we define a relation as follows. If F,F𝒟~m,l, then FF if and only if for any open subgroup HGKm containing GK[m,m-1], the images of FH and FH in H1=Hab are commensurable. It is easy to see that is an equivalence relation on the set 𝒟~m,l, and we write 𝒟m,l=def𝒟~m,l/ for the corresponding set of equivalence classes.

Proposition 23.

We use the above notations. Then there exists a natural map

ϕm,l:𝒟m,l𝔓𝔯𝔦𝔪𝔢𝔰Km-1na

with the following properties.

1. The map ϕm,l is injective, and induces a bijection

𝒟m,l𝔓𝔯𝔦𝔪𝔢𝔰Km-1na,(l),

where 𝔓𝔯𝔦𝔪𝔢𝔰Km-1na,(l)𝔓𝔯𝔦𝔪𝔢𝔰Km-1na denotes the set of nonarchimedean primes of Km-1 whose image in 𝔓𝔯𝔦𝔪𝔢𝔰 is distinct from l.

2. The map ϕm,l is GKm-equivariant with respect to the natural actions of GKm on 𝒟m,l and 𝔓𝔯𝔦𝔪𝔢𝔰Km-1na. In particular, the action of GKm on 𝒟m,l factors through GKm-1.

3. Let a𝒟m,l and 𝔭¯=defϕm,l(a). Then the stabiliser

St(a)=def{gGKm-1ga=a}

coincides with the decomposition group D𝔭¯GKm-1 at 𝔭¯.

Proof.

First, Proposition 22 implies that there exists a well-defined surjection

ϕ~m,l:𝒟~m,l𝔓𝔯𝔦𝔪𝔢𝔰Km-1na,(l),F𝔭¯

in the notation of Proposition 22. We prove that ϕ~m,l factors as

𝒟~m,l𝒟m,lϕm,l𝔓𝔯𝔦𝔪𝔢𝔰Km-1na.

Indeed, let F,F𝒟~m,l; assume FF. By Proposition 22, there are 𝔭,𝔭𝔓𝔯𝔦𝔪𝔢𝔰Kmna,(l) such that F and F are open subgroups of l-Sylow subgroups of D𝔭 and D𝔭, respectively. Let M/K be any finite subextension of Km-1/K, which corresponds to an open subgroup HGKm containing GK[m,m-1]. Write 𝔭¯,𝔭¯𝔓𝔯𝔦𝔪𝔢𝔰Km-1na and 𝔭M,𝔭M𝔓𝔯𝔦𝔪𝔢𝔰Mna for the images of 𝔭,𝔭, respectively. As FF, the images of FH and FH in H1=GM1 are commensurable, hence the decomposition groups

D𝔭M1,D𝔭M1GM1

are l-commensurable. In particular, the intersection D𝔭M1D𝔭M1 is l-infinite, hence nontrivial (cf. Proposition 1 (v)). Now, by Proposition 3, (i)  (ii), one has 𝔭M=𝔭M. As M/K is an arbitrary finite subextension of Km-1/K, this shows 𝔭¯=𝔭¯, as desired.

We prove assertion (i). Assume F,F𝒟~m,l have the same image 𝔭¯𝔓𝔯𝔦𝔪𝔢𝔰Km-1na,(l) via the above map 𝒟~m,l𝔓𝔯𝔦𝔪𝔢𝔰Km-1na. This means that there exist 𝔭,𝔭𝔓𝔯𝔦𝔪𝔢𝔰Kmna above the prime 𝔭¯ (which is not above the prime l𝔓𝔯𝔦𝔪𝔢𝔰), with decomposition groups

D𝔭,D𝔭GKm

such that F,F are l-open subgroups of D𝔭,D𝔭, respectively (cf. Proposition 22). We show FF. Let HGKm be any open subgroup containing GK[m,m-1], and

M=defKmHKm-1.

Let 𝔭M be the image of 𝔭¯ in M. The images of FH and FH in H1=GM1 are both l-open in the decomposition group D𝔭MGM1=GMab at 𝔭M, hence both open in the l-Sylow subgroup D𝔭M,l of D𝔭M. Thus, they are commensurable, and FF. The second assertion in (i) follows by considering for a nonarchimedean prime 𝔭¯𝔓𝔯𝔦𝔪𝔢𝔰Km-1na of residue characteristic l, a prime 𝔭𝔓𝔯𝔦𝔪𝔢𝔰Kmna above 𝔭¯, and an l-Sylow subgroup of the decomposition group D𝔭GKm at 𝔭, which satisfies condition (l).

Further, GKm acts naturally on 𝒟~m,l via the action on its subgroups by conjugation and the map 𝒟~m,l𝔓𝔯𝔦𝔪𝔢𝔰Km-1na,(l) is clearly GKm-equivariant. Assertion (ii) follows from this, together with assertion (i). Assertion (iii) follows from assertions (i) and (ii). ∎

Let m2 be an integer. For each prime number l set (cf. Proposition 23 (iii))

St(𝒟m,l)=def{St(a)a𝒟m,l}Sub(GKm-1),

and set

St(𝒟m)=deflSt(𝒟m,l),

where the union is over all prime numbers l.

Proposition 24.

We use the above notations. Let m2 be an integer, and l1l2 prime numbers. Then one has

Dec(Km-1/K)=St(𝒟m)=St(𝒟m,l1)St(𝒟m,l2)

in Sub(GKm-1). In particular, the subset Dec(Km-1/K)Sub(GKm-1) can be recovered group-theoretically from GKm+1.

Proof.

The first assertion follows from the various definitions and Proposition 23. The second assertion follows from the first and the fact that for each prime number l, the GKm-1-set 𝒟m,l, hence the subset St(𝒟m,l)Sub(GKm-1), can be recovered group-theoretically from GKm+1 (cf. Remark 21 (ii)). ∎

We have a natural surjective map

The map dm-1 is bijective if m3 (cf. Corollary 6). Further, if m=2, then the surjective map

induces a natural bijective map

𝔓𝔯𝔦𝔪𝔢𝔰Kna𝑑Dec(K1/K)

(cf. Proposition 3). In summary, as a consequence of Proposition 24, we have the following. (Here we make a renumbering by replacing m-1 with m.)

Theorem 25.

Let m2 (resp. m=1) be an integer. The GKm-set Dec(Km/K) (resp. the set Dec(K1/K)), hence the GKm-set PrimesKmna (resp. the set PrimesKna), can be recovered group-theoretically from GKm+2.

For an integer m1, let χcycl:GKm(GK1)^× be the cyclotomic character.

Theorem 26.

Let m3 be an integer. Then χcycl can be recovered group-theoretically from GKm.

Proof.

We may assume m=3, and show that for each prime number l, the l-part of the cyclotomic character χcycl(l):GK2l× can be recovered group-theoretically from GK3. We claim that the following group-theoretic characterisation of χcycl(l) holds: Let χ:GK2l× be a character. Then one has χ=χcycl(l) if and only if for every F𝒟~2,l, every hFGK[2,1] and every gNGK2(FGK[2,1]), one has

ghg-1=hχ(g).

(Recall that the definition of the set 𝒟~2,l(Sub(GK2)) involves GK3.)

To prove this claim, we first determine the normaliser NGK2(FGK[2,1]) for each F𝒟~2,l. By Proposition 22, F is an open subgroup of an l-Sylow subgroup of the decomposition group D𝔭GK2 at some 𝔭𝔓𝔯𝔦𝔪𝔢𝔰K2na with residue characteristic l, and the image 𝔭¯𝔓𝔯𝔦𝔪𝔢𝔰K1na of 𝔭 is uniquely determined by F. Then one has

NGK2(FGK[2,1])=D𝔭GK[2,1]=π-1(D𝔭¯),

where π denotes the projection GK2GK1. Indeed, write D𝔭I𝔭I𝔭tame for the inertia and the tame inertia groups. By Proposition 1 (v), D𝔭GK[2,1]I𝔭, hence

FGK[2,1](D𝔭GK[2,1])l(D𝔭GK[2,1])(l)I𝔭(l)(I𝔭tame)(l)l,

where (D𝔭GK[2,1])l stands for the l-Sylow subgroup of the profinite abelian group D𝔭GK[2,1], and the last isomorphism follows from Proposition 1 (vi). As F is an open subgroup of an l-Sylow subgroup of the decomposition group D𝔭GK2, FGK[2,1] is open in (D𝔭GK[2,1])l. By Proposition 1 (i),

D𝔭¯GK𝔭0ab,

where 𝔭0 is the image of 𝔭 in 𝔓𝔯𝔦𝔪𝔢𝔰Kna, hence, in particular, (by local class field theory) the l-Sylow group of D𝔭¯ is isomorphic to the direct product of a finite cyclic group of l-power order (that is the l-Sylow subgroup of κ(𝔭0)×) and l. It follows from this that the inclusion (D𝔭GK[2,1])(l)I𝔭(l)(l) is open. Now, since D𝔭GK[2,1](=Ker(D𝔭GK1) is normal in D𝔭 and FGK[2,1]=((D𝔭GK[2,1])l)((D𝔭GK[2,1])l:FGK[2,1]) is characteristic in D𝔭GK[2,1], FGK[2,1] is normal in D𝔭, hence D𝔭NGK2(FGK[2,1]). On the other hand, since GK[2,1](=GK[1]ab) is abelian, one has

GK[2,1]NGK2(FGK[2,1]).

Thus, D𝔭GK[2,1]NGK2(FGK[2,1]). Conversely, let gNGK2(FGK[2,1]). Then

D𝔭FGK[2,1]=g(FGK[2,1])g-1gD𝔭g-1=Dg𝔭.

As the inclusion FGK[2,1]I𝔭(l)(l) is open, FGK[2,1] is nontrivial. Therefore, by Proposition 3, (i)  (ii), one has 𝔭¯=g𝔭¯=π(g)𝔭¯, which implies π(g)D𝔭¯. Thus,

gπ-1(D𝔭¯)=D𝔭GK[2,1].

As gNGK2(FGK[2,1]) is arbitrary, we conclude

NGK2(FGK[2,1])D𝔭GK[2,1],

hence NGK2(FGK[2,1])=D𝔭GK[2,1].

Now, we prove the “only if” part of the claim. As the inclusion FGK[2,1]I𝔭(l) is compatible with the actions of D𝔭 (by conjugation), one has

ghg-1=hχcycl(l)(g)

for every hFGK[2,1] and every gD𝔭. On the other hand, as GK[2,1] is abelian and χcycl(l):GK2l× factors through π:GK2GK1, one has

ghg-1=h=hχcycl(l)(g)

for every hFGK[2,1] and every gGK[2,1]. Thus,

ghg-1=hχcycl(l)(g)

for every hFGK[2,1] and every gD𝔭GK[2,1]=NGK2(FGK[2,1]), as desired.

Finally, we prove the “if” part of the claim. For this, it suffices to show that GK2 is (topologically) generated by NGK2(FGK[2,1]) (F𝒟~2,l). As in the preceding arguments, one has FD𝔭 for some 𝔭𝔓𝔯𝔦𝔪𝔢𝔰K2na and NGK2(FGK[2,1])=D𝔭GK[2,1]. So, it suffices to prove that GK1 is generated by D𝔭¯, where 𝔭¯ is the image of 𝔭 in 𝔓𝔯𝔦𝔪𝔢𝔰K1. This follows from Chebotarev’s density theorem, together with (the surjectivity in) Proposition 23 (i), as desired. ∎

Corollary 27.

Let m2 (resp. m=1) be an integer, let K,L be number fields, and let τm+2:GKm+2GLm+2 be an isomorphism of profinite groups. Let τm:GKmGLm be the isomorphism of profinite groups induced by τm+2.

1. There exists a unique bijection

ϕm:𝔓𝔯𝔦𝔪𝔢𝔰Kmna𝔓𝔯𝔦𝔪𝔢𝔰Lmna(resp.ϕ:𝔓𝔯𝔦𝔪𝔢𝔰Kna𝔓𝔯𝔦𝔪𝔢𝔰Lna)

such that the diagram commutes

(resp.

where τ~m is induced by τm:GKmGLm (i.e. τ~m(D)=defτm(D)), the vertical maps are the natural (bijective) ones (cf. the paragraph before Theorem 25). For m2, ϕm is compatible with τm and the natural actions of GKm, GLm on 𝔓𝔯𝔦𝔪𝔢𝔰Kmna, 𝔓𝔯𝔦𝔪𝔢𝔰Lmna, respectively, hence, in particular, ϕm induces a unique bijection ϕ:𝔓𝔯𝔦𝔪𝔢𝔰Kna𝔓𝔯𝔦𝔪𝔢𝔰Lna.

2. The bijection ϕ:𝔓𝔯𝔦𝔪𝔢𝔰Kna𝔓𝔯𝔦𝔪𝔢𝔰Lna fits into the following commutative diagram:

where the vertical maps are the natural ones (𝔭(the characteristic of κ(𝔭))).

3. Let 𝔭𝔓𝔯𝔦𝔪𝔢𝔰Kna and 𝔮=defϕ(𝔭)𝔓𝔯𝔦𝔪𝔢𝔰Lna. Then d𝔭=d𝔮, e𝔭=e𝔮, f𝔭=f𝔮, and N(𝔭)=N(𝔮). In particular, K and L are arithmetically equivalent (cf. Notations).

4. Let 𝔭𝔓𝔯𝔦𝔪𝔢𝔰Kmna (resp. let 𝔭𝔓𝔯𝔦𝔪𝔢𝔰Kna) and let 𝔮=defϕm(𝔭)𝔓𝔯𝔦𝔪𝔢𝔰Lmna (resp. let 𝔮=defϕ(𝔭)𝔓𝔯𝔦𝔪𝔢𝔰Lna). Let D𝔭,D𝔮 be the decomposition subgroups of GKm,GLm, respectively, corresponding to 𝔭,𝔮, respectively, I𝔭,I𝔮 the inertia subgroups of D𝔭,D𝔮, respectively, and Frob𝔭D𝔭/I𝔭,Frob𝔮D𝔮/I𝔮 the Frobenius elements. Then the isomorphism D𝔭D𝔮 induced by τm (cf. part (i)) restricts to an isomorphism I𝔭I𝔮. Further, the induced isomorphism D𝔭/I𝔭D𝔮/I𝔮 maps Frob𝔭 to Frob𝔮.

5. Assume m2. Let H be an open subgroup of GKm and K/K (resp. L/L) the finite subextension of Km/K (resp. Lm/L) corresponding to HGKm (resp. τm(H)GLm). Let ϕ:𝔓𝔯𝔦𝔪𝔢𝔰Kna𝔓𝔯𝔦𝔪𝔢𝔰Lna be the bijection induced by the bijection

ϕm:𝔓𝔯𝔦𝔪𝔢𝔰Kmna𝔓𝔯𝔦𝔪𝔢𝔰Lmna

(cf. part (i)). Moreover, let 𝔭𝔓𝔯𝔦𝔪𝔢𝔰Kna and 𝔮=defϕ(𝔭)𝔓𝔯𝔦𝔪𝔢𝔰Lna. Then d𝔭=d𝔮, e𝔭=e𝔮, f𝔭=f𝔮, and N(𝔭)=N(𝔮). In particular, K and L are arithmetically equivalent.

Proof.

(i) This is a precise reformulation of Theorem 25, hence follows from Proposition 24, together with Corollary 6 and Proposition 3.

(ii) This follows from (i) and Proposition 1 (ii).

(iii) This follows from (i), (ii) and Proposition 1 (ii)–(iv).

(iv) By (ii) and Theorem 26, the following diagram commutes:

where p is the characteristic of κ(𝔭) and χcycl(p) stands for the prime-to-p part of the cyclotomic character χcycl. Now, since

Ker(χcycl(p)|D𝔭)=I𝔭andKer(χcycl(p)|D𝔮)=I𝔮,

the first assertion follows, and χcycl(p)|D𝔭 (resp. χcycl(p)|D𝔮) induces an injective map

D𝔭/I𝔭(^(p))×(resp. D𝔮/I𝔮(^(p))×),

which is again denoted by χcycl(p). Since Frob𝔭D𝔭/I𝔭 (resp. Frob𝔮D𝔮/I𝔮) is characterised by

χcycl(p)(Frob𝔭)=pf𝔭(resp. χcycl(p)(Frob𝔮)=pf𝔮),

the second assertion follows from (ii) and (iii).

(v) Write 𝔭 (resp. p) for the image of 𝔭 in 𝔓𝔯𝔦𝔪𝔢𝔰Kna (resp. 𝔓𝔯𝔦𝔪𝔢𝔰na) and set

𝔮=ϕ(𝔭)𝔓𝔯𝔦𝔪𝔢𝔰Lna.

Take 𝔭~𝔓𝔯𝔦𝔪𝔢𝔰Kmna above 𝔭 and set 𝔮~=defϕm(𝔭~)𝔓𝔯𝔦𝔪𝔢𝔰Lmna. Then 𝔮 and p (resp. 𝔮~) are below (resp. is above) 𝔮 (cf. (i) and (ii)). By (i) and (iv), one has

τm:D𝔭~D𝔮~andτm:I𝔭~I𝔮~,

hence

d𝔭=(D𝔭~:D𝔭~H)d𝔭=(D𝔮~:D𝔮~τm(H))d𝔮=d𝔮,
e𝔭=(I𝔭~:I𝔭~H)e𝔭=(I𝔮~:I𝔮~τm(H))e𝔮=e𝔮,
f𝔭=d𝔭/e𝔭=d𝔮/e𝔮=f𝔮,
N(𝔭)=pf𝔭=pf𝔮=N(𝔮),

where the first and the second formulae follow from (iii), the third formula follows form the first and the second, and the last formula follows from the third. This finishes the proof of Corollary 27. ∎

2 Proof of Theorem 1

For a number field K let IK be the multiplicative monoid freely generated by the elements of 𝔓𝔯𝔦𝔪𝔢𝔰Kna (cf. [2]), endowed with the norm function defined by 𝔭N(𝔭) (𝔭𝔓𝔯𝔦𝔪𝔢𝔰Kna). Let K,L be number fields and τ:GK3GL3 an isomorphism of profinite groups. We show K and L are isomorphic. The bijection ϕ:𝔓𝔯𝔦𝔪𝔢𝔰Kna𝔓𝔯𝔦𝔪𝔢𝔰Lna in Corollary 27 (i) induces a norm-preserving bijection ϕ~:IKIL (cf. Corollary 27 (iii)). Let NGK1 be an open subgroup corresponding to the extension K/K with K=def(K1)N, and M=defτ1(N) which corresponds to the extension L/L with L=def(L1)M. Let 𝔭𝔓𝔯𝔦𝔪𝔢𝔰Kna and 𝔮=defϕ(𝔭). Then 𝔭 (resp. 𝔮) is unramified in the extension K/K (resp. L/L) if and only if the image of I𝔭 (resp. I𝔮) in Gal(K/K) (resp. Gal(L/L)) is trivial. In particular, 𝔭 is unramified in K/K if and only if 𝔮 is unramified in L/L, and the isomorphism Gal(K/K)Gal(L/L) induced by τ1:GK1GL1 maps the image of Frob𝔭 to that of Frob𝔮 (cf. Corollary 27 (iv)). Thus, by the main theorem of [2], there exists a field isomorphism σ:KL (cf. [2, Theorem 3.1], the equivalence (ii)  (iv)). This finishes the proof of Theorem 1.

3 Proof of Theorem 2

Let m0 be an integer, K,L number fields, and τm+3:GKm+3GLm+3 an isomorphism of profinite groups. We show the existence of a field isomorphism σm:KmLm such that τm(g)=σmgσm-1 for every gGKm, where τm:GKmGLm is the isomorphism induced by τm+3. We follow Uchida’s method in [13]. Let K/K be a finite Galois subextension of Km/K with Galois group H, corresponding to a normal open subgroup UGKm, and V=defτm(U) corresponding to a finite Galois subextension L/L of Lm/L with Galois group J. The isomorphism τm induces naturally an isomorphism τ:HJ. Let T(K) be the (finite) set of field isomorphisms σ:KL such that τ(h)=σhσ-1 for every hH. It is easy to see that {T(K)}K/K forms a projective system, and the projective limit limK/KT(K) consists of isomorphisms σm:KmLm satisfying the condition in Theorem 2. Further, if T(K) for any K as above, then the projective limit limK/KT(K) over all such finite sets T(K) would be nonempty.

Now, the same proof as in [13] shows that T(K). More precisely, the proof in loc. cit. applies as it is by noting the following. First, one needs to know that K and L above are arithmetically equivalent, which in our case follows from Corollary 27 (v). Second, one needs to know that certain finite abelian extensions of K and L introduced (and denoted by j=0mM1,j and j=0mM2,j) in [13] are arithmetically equivalent. Since these extensions are contained in Km+1 and Lm+1, respectively, and correspond to each other via τm+1:GKm+1GLm+1, this follows again from Corollary 27 (v) (applied to m+1 and m+3=(m+1)+2 instead of m and m+2). The rest of the proof that T(K) is the same as in loc. cit. This finishes the proof of (i).

Next, assume that m1 and let σm,i:KmLm be isomorphisms for i=1,2 such that τm(g)=σm,igσm,i-1 for i=1,2. In particular, for each index i=1,2, the isomorphism σm,i:KmLm induces an isomorphism σi:KL. Also, for each i=1,2, the isomorphism σm,i:KmLm induces a bijection ϕm,i:𝔓𝔯𝔦𝔪𝔢𝔰(Km)na𝔓𝔯𝔦𝔪𝔢𝔰(Lm)na, and, for every 𝔭𝔓𝔯𝔦𝔪𝔢𝔰(Km)na, one has

Dϕm,i(𝔭)=σm,iD𝔭σm,i-1=τm(D𝔭).

Thus, if m2 (resp. m=1), the bijections ϕm,i:𝔓𝔯𝔦𝔪𝔢𝔰(Km)na𝔓𝔯𝔦𝔪𝔢𝔰(Lm)na (resp. ϕi:𝔓𝔯𝔦𝔪𝔢𝔰(K)na𝔓𝔯𝔦𝔪𝔢𝔰(L)na induced by σi:KL) for i=1,2 must coincide with each other (cf. Corollary 6 (resp. Proposition 3)). This, together with Lemma 8, finishes the proof of (ii). This finishes the proof of Theorem 2.

To the memory of Professor Michel Raynaud

A Recovering the cyclotomic character from GK2

In this appendix we use the notations in the Introduction. Fix a prime number l, and set l~=defl (resp. 4) for l2 (resp. l=2) (cf. Notations). In Theorem 26, we proved that the cyclotomic character χcycl (hence its l-part χcycl(l)) can be recovered group-theoretically from GKm, if m3. In this appendix, we will prove that χcycl(l) can be recovered group-theoretically from GK2, up to twists by finite characters.

For an integer m1 let GKmΓ be the maximal quotient of GKm which is pro-l abelian and torsion free. Thus, Γ (depends only on GK1 and) is (non-canonically) isomorphic to lr for some integer r with r2+1r[K:], where r2 is the number of complex primes of K; Leopoldt’s conjecture predicts the equality r=r2+1 (cf. [9, Chapter X, Section 3]). Write K()/K for the corresponding (infinite) subextension of K¯/K with Galois group Γ. (Note that K()K1.) Write K(n)/K for the (finite) subextension of K()/K corresponding to the subgroup Γ(n)=def{γlnγΓ}Γ, n0 an integer. We denote by Λ=l[[Γ]] the associated complete group ring (cf. [9, Chapter V, Section 2]). It is known that given a set of free generators {γ1,,γr} of Γ, we have an isomorphism Λl[[T1,,Tr]], γi1+Ti (1ir). (See [9, (5.3.5) Proposition] in case r=1. The general case is similar.) Slightly more generally, let 𝒪/l be a finite extension of (complete) discrete valuation rings. Then we denote by Λ𝒪=𝒪[[Γ]]=Λl𝒪 the associated complete group ring over 𝒪 (cf. [9]). Consider the exact sequence 1HGK2Γ0 where H=defKer(GK2Γ). By pushing out this sequence by the projection HP=def(H(l))ab, we obtain an exact sequence 1PQΓ1. Write K/K for the corresponding subextension of K¯/K with Galois group Q. (Note that KK2.) Thus, K/K() is the maximal abelian pro-l extension of K(). Note that the quotient GK2Q can be reconstructed group-theoretically from GK2 by its very definition. Let Σ=def{l,}(K) be the finite set of primes of K consisting of the nonarchimedean primes above l and the primes at infinity, and SΣ a finite set of primes of K. Write PPΣ (resp. PPS) for the quotient corresponding to the maximal subextension K(),Σ/K() (resp. K(),S/K()) of K/K() which is unramified outside Σ(K()) (resp. S(K())). Thus, K=SK(),S, where the union is over all finite sets S of primes of K containing Σ. Note that PS (resp. P) has a natural structure of Λ-module of which Ker(PSPΣ) (resp. Ker(PPΣ)) is a Λ-submodule. For a (nonarchimedean) prime 𝔭𝔓𝔯𝔦𝔪𝔢𝔰KΣ of K we write Γ𝔭 for the decomposition group of Γ at 𝔭 (which is well defined since Γ is abelian). Then Γ𝔭 is canonically generated by the Frobenius element γ𝔭 at 𝔭 (i.e. the image of Frob𝔭 in Γ) and isomorphic to l. Note that no nonarchimedean prime of K splits completely in K(), and the extension K()/K is unramified outside Σ. Let χcycl(l):GK1l× be the l-part of the cyclotomic character. Then χcycl(l)¯ (cf. Notations) factors as

GK1Γ𝑤1+l~l.

We will refer to the induced character w:Γ1+l~ll× as the cyclotomic character of Γ. Thus, the goal of this section is to show that w can be recovered group-theoretically from GK2 (cf. Proposition 9).

Proposition 1.

Let SΣ be a finite set of primes of K. Then there exists a canonical exact sequence of Λ-modules

0𝔭SΣμlκ(𝔭)IndΓΓ𝔭l(1)PSPΣ0

(where for an integer N>0, μN stands for the group of N-th roots of unity), and passing to the projective limit over all finite sets SΣ we obtain a canonical exact sequence of Λ-modules

(A.1)0𝔭𝔓𝔯𝔦𝔪𝔢𝔰KΣμlκ(𝔭)IndΓΓ𝔭l(1)PPΣ0.

Further, for each pPrimesKΣ with μlκ(p), the Λ-module IndΓΓpZl(1) is isomorphic to Λ/γp-ϵpw(γp), where ϵp=1 (resp. ϵp=-1) if μl~κ(p) (resp. μl~κ(p)). (In particular, ϵp=1 if l2.)

Proof.

We follow the arguments in [9, proof of (11.3.5) Theorem]. First, the weak Leopoldt conjecture holds for the extension K()/K. In other words, let KS/K (resp. KΣ/K) be the maximal subextension of K¯/K which is unramified outside S (resp. Σ), then (cf. [10, Corollaire 2.9])

H2(Gal(KΣ/K()),l/l)=0.

Thus, we have an exact sequence of cohomology groups with l/l-coefficients

0H1(Gal(KΣ/K()))H1(Gal(KS/K()))
H1(Gal(KS/KΣ))Gal(KΣ/K())0,

or dually, using [9, (10.5.4) Corollary],

0limn𝔭(n)SΣ(K(n))(I𝔭(n)(l))G(K(n))𝔭(n)PSPΣ0,

where I𝔭(n) is the inertia subgroup in G(K(n))𝔭(n), and (I𝔭(n)(l))G(K(n))𝔭(n) is the coinvariant of the G(K(n))𝔭(n)-module I𝔭(n)(l). Further,

limn𝔭(n)SΣ(K(n))(I𝔭(n)(l))G(K(n))𝔭(n)=limn𝔭SΣIndGal(K(n)/K)Gal((K(n))𝔭(n)/K𝔭)(I𝔭(l))G(K(n))𝔭(n)
=𝔭SΣlimnIndGal(K(n)/K)Gal((K(n))𝔭(n)/K𝔭)(I𝔭(l))G(K(n))𝔭(n)
=𝔭SΣIndΓΓ𝔭(I𝔭(l))G(K())𝔭()

where in the second and third terms (resp. in the fourth term) 𝔭(n) (resp. 𝔭()) stands for a fixed prime in 𝔓𝔯𝔦𝔪𝔢𝔰K(n) (resp. 𝔓𝔯𝔦𝔪𝔢𝔰K()) above 𝔭. It follows from the various definitions (cf. [9, proof of (11.3.5) Theorem]) that one has

(I𝔭(l))G(K())𝔭()={0,μlκ(𝔭),I𝔭(l)l(1),μlκ(𝔭).

Thus, the first exact sequence in Proposition 1 is obtained, and the exact sequence (A.1) is obtained by passing to the projective limit. The last assertion follows immediately from the various definitions. (Observe that χcycl(l)(Frob𝔭)=ϵ𝔭w(γ𝔭) holds for 𝔭 with μlκ(𝔭).) ∎

For 𝔭𝔓𝔯𝔦𝔪𝔢𝔰KΣ with μlκ(𝔭), we write

J𝔭=defIndΓΓ𝔭l(1)andJ=def𝔭𝔓𝔯𝔦𝔪𝔢𝔰KΣμlκ(𝔭)J𝔭.

Thus, we have the exact sequence

(A.2)0JPPΣ0.

Write

Γprim=defΓΓ(1),

and let {γ1,,γr} be a set of free generators of Γ. Let γΓ{1}. Then one may write

γ=i=1rγiαi,γ

with αi,γl. Write Γγ for the subgroup γ of Γ (topologically) generated by γ. Then (Γ/Γγ)tor is finite cyclic of order lmγ for some mγ0. There exists a unique element γ~Γprim such that γ~lmγ=γ. For 𝔭𝔓𝔯𝔦𝔪𝔢𝔰KΣ, we write αi,𝔭 and m𝔭 instead of αi,γ𝔭 and mγ𝔭, respectively. The following lemma will be useful.

Lemma 2.

With the above notations, the following (i)–(vii) are equivalent:

1. (Γ/Γγ)tor=0 (i.e. mγ=0).

2. γ is a member of a set of free generators of Γ.

3. The image of γ under the map ΓΓl𝔽l is nontrivial.

4. αi,γl× for some 1ir.

5. γΓprim.

6. For every α1+ll, γ-α is a prime element of Λ.

7. For every finite extension 𝒪/l of discrete valuation rings and every α1+𝔪, where 𝔪 is the maximal ideal of 𝒪, γ-α is a prime element of Λ𝒪.

Easy. ∎

Lemma 3.

Let O/Zl be a finite extension of (complete) discrete valuation rings and mO the maximal ideal. Let γΓ, γΓprim, and α,α1+m. Then γ-α divides γ-α in ΛO (i.e. γ-αγ-αΛO) if and only if there exists νZl such that γ=(γ)ν and α=(α)ν. In particular, (γ-α)(γ-α) implies γγ.

Proof.

First, suppose that there exists νl such that γ=(γ)ν and α=(α)ν. Then

γ-α=(γ)ν-(α)ν=i=0(νi)(γ-1)i-i=0(νi)(α-1)i
=i=0(νi)((γ-1)i-(α-1)i)
=(γ-α)i=1(νi)j=0i-1(γ-1)j(α-1)i-1-j

is divisible by γ-α.

Next, suppose (γ-α)(γ-α). By Lemma 2, there exists a set of free generators {γ1,γr} of Γ with γ1=γ. Let Γ be the closed subgroup of Γ generated by {γ2,,γr}, and set Λ𝒪=def𝒪[[Γ]]. Consider the surjective homomorphism Λ𝒪Λ𝒪 of 𝒪-algebras, defined by γ1=γα1+𝔪𝒪×(Λ𝒪)× and γiγi (i=2,,r). Following the decomposition Γ(γ)l×Γ, write γ=(γ)νγ, where νl,γΓ. Then the image of γ-α in Λ𝒪 is (α)νγ-α, which must be 0 by assumption. Thus, γ=(α)-να𝒪 in Λ𝒪, which first implies γ=1 (i.e. γ=(γ)ν) and then (α)-να=1 (i.e. α=(α)ν), as desired. ∎

The cyclotomic character w:Γ1+l~l induces naturally a continuous surjective homomorphism of l-algebras ψw:Λ=l[[Γ]]l such that ψw(γ)=w(γ) if γΓ.

Definition 4.

Let M be a Λ-module. We define

IM=defγ-α¯γΓ,α1+ll, and AnnΛ(x)=γ-αΛ for some xM{0}

which is an ideal of Λ. (For the map l×1+l~l, αα¯, see Notations.) Note that if MM, then IMIM. Further, IM=IMΛ-tor.

Of particular interest to us is the ideal IJ of Λ.

Lemma 5.

Let {γ1,,γr} be a set of free generators of Γ. Consider the following ideals of Λ:

I~=defKer(ψw),
I=defγ-w(γ)γΓ,
I=Iγ1,,γr=defγ1-w(γ1),,γr-w(γr).

Then the following equalities hold:

IJ=I~=I=I.

(In particular, I does not depend on the choice of {γ1,,γr}.)

Proof.

The inclusions III~ clearly hold for any {γ1,,γr}. We show I~I for any {γ1,,γr}. As II~, we have a surjective homomorphism Λ/IΛ/I~=l. Using the isomorphism Λl[[T1,,Tr]], γi1+Ti (1ir), one sees easily that Λ/Il, hence the homomorphism Λ/IΛ/I~ is an isomorphism. Thus, the equalities I=I=I~ follow. (In particular, I does not depend on the choice of {γ1,,γr}.)

Next, we prove IJI. For each 𝔭𝔓𝔯𝔦𝔪𝔢𝔰KΣ with μlκ(𝔭), take an element γ~𝔭Γ as in the paragraph preceding Lemma 2. Then, by Proposition 1,

J𝔭=IndΓΓ𝔭l(1)Λ/γ𝔭-ϵ𝔭w(γ𝔭)Λ=Λ/γ~𝔭lm𝔭-ϵ𝔭w(γ~𝔭)lm𝔭Λ.

Further, write E𝔭 for the set of lm𝔭-th roots of ϵ𝔭 in ¯l, and set

𝒪E𝔭=defl[E𝔭]¯landΛE𝔭=defΛ𝒪E𝔭.

More concretely, if ϵ𝔭=1 (resp. ϵ𝔭=-1), then E𝔭=μlm𝔭 (resp. E𝔭=μ2m𝔭+1μ2m𝔭) and 𝒪E𝔭=l[ζ], where ζ is a primitive lm𝔭-th (resp. 2m𝔭+1-th) root of unity in ¯l. Now, one has

J𝔭J𝔭l𝒪E𝔭ΛE𝔭/γ~𝔭lm𝔭-ϵ𝔭w(γ~𝔭)lm𝔭ΛE𝔭ηE𝔭ΛE𝔭/γ~𝔭-ηw(γ~𝔭)ΛE𝔭.

Here, the first injection comes from the fact that 𝒪E𝔭 is free (of rank >0) as a l-module, while the second injection comes from the fact that ΛE𝔭 (𝒪E𝔭[[T1,,Tr]]) is a unique factorisation domain. Set J𝔭,η=defΛE𝔭/γ~𝔭-ηw(γ~𝔭)ΛE𝔭. Thus, one has, as Λ-modules,

J=𝔭J𝔭𝔭ηJ𝔭,η.

Let xJ{0} such that AnnΛ(x)=γ-αΛIJ with γΓ and α1+ll. Via the above injections, x is identified with (x𝔭,η)𝔭,η, where x𝔭,ηJ𝔭,η for 𝔭𝔓𝔯𝔦𝔪𝔢𝔰KΣ with μlκ(𝔭) and ηE𝔭. As x0, there exists (𝔭,η) such that x𝔭,η0, and

γ-αAnnΛ(x)AnnΛE𝔭(x𝔭,η)=γ~𝔭-ηw(γ~𝔭)ΛE𝔭,

where the equality follows from the fact that ΛE𝔭/γ~𝔭-ηw(γ~𝔭)ΛE𝔭 is an integral domain (cf. Lemma 2, (v)  (vii)). Now, by Lemma 3, there exists νl such that γ=γ~𝔭ν and α=(ηw(γ~𝔭))ν=ηνw(γ~𝔭)ν. Further, one has

αw(γ~𝔭)-ν=ηνl×(OE𝔭×)tor=(l×)tor.

As αw(γ~𝔭)-ν(l×)tor, one has

α¯=w(γ~𝔭)ν¯=w(γ~𝔭)ν.

Thus, γ-α¯=γ~𝔭ν-w(γ~𝔭)ν=γ~𝔭ν-w(γ~𝔭ν)I, as desired.

Finally, we prove that I=Iγ1,,γrIJ (for some choice of a set of free generators {γ1,,γr} of Γ). Let {v1,,vr} be a basis of Γl𝔽l. Then, by Chebotarev’s density theorem there exists {𝔭1,,𝔭r}𝔓𝔯𝔦𝔪𝔢𝔰KΣ such that μlκ(𝔭i), 1ir, and that the Frobenius element γ𝔭i at 𝔭i maps to viΓl𝔽l, 1ir. By Nakayama’s lemma, {γ𝔭1,,γ𝔭r} is a set of free generators of Γ. Fix any i{1,,r}. By Proposition 1, we have J𝔭iΛ/γ𝔭i-ϵ𝔭iw(γ𝔭i) as Λ-modules. Let t𝔭i be the element of J𝔭i(J) corresponding to 1Λ/γ𝔭i-ϵ𝔭iw(γ𝔭i) under this isomorphism. Then

AnnΛ(t𝔭i)=γ𝔭i-ϵ𝔭iw(γ𝔭i)Λ,

hence by definition, γ𝔭i-w(γ𝔭i)=γ𝔭i-ϵ𝔭iw(γ𝔭i)¯IJ. As i{1,,r} is arbitrary, this shows I=Iγ𝔭1,,γ𝔭rIJ, as desired. This finishes the proof of Lemma 5. ∎

Lemma 6.

Assume r2 and let fΛ{0}. Then IfJ=IJ.

Proof.

Since fJJ, one has IfJIJ. So, we show the converse IJIfJ. As in the last part of the proof of Lemma 5, let {v1,,vr} be a basis of Γl𝔽l, and take {𝔭1,,𝔭r}𝔓𝔯𝔦𝔪𝔢𝔰KΣ such that μlκ(𝔭i), 1ir, and that γ𝔭iΓ maps to viΓl𝔽l, 1ir. Thus, {γ𝔭1,,γ𝔭r} is a set of free generators of Γ, and

IJ=I=γ𝔭i-w(γ𝔭i),1irΛ

(cf. Lemma 5). Further,

J𝔭iΛ/γ𝔭i-ϵ𝔭iw(γ𝔭i)

as Λ-modules; Λ/γ𝔭i-ϵ𝔭iw(γ𝔭i) is an integral domain (cf. Lemma 2, (ii)  (vi)); and fJ𝔭ifJ. Accordingly, if fγ𝔭i-ϵ𝔭iw(γ𝔭i) and t𝔭i is the element of J𝔭i(J) corresponding to 1Λ/γ𝔭i-ϵ𝔭iw(γ𝔭i), then

AnnΛ(ft𝔭i)=γ𝔭i-ϵ𝔭iw(γ𝔭i)Λ,

hence

γ𝔭i-w(γ𝔭i)=γ𝔭i-ϵ𝔭iw(γ𝔭i)¯IfJ.

Thus, to prove IJ(=Iγ𝔭1,,γ𝔭r)IfJ, it suffices to show that given fΛ{0}, we can choose {𝔭1,,𝔭r} as above such that (γ𝔭i-ϵ𝔭iw(γ𝔭i))f for 1ir. Further, (as Λl[[T1,,Tr]] is a unique factorisation domain) this would follow if one shows that for each i{1,,r}, there exists an infinite subset {𝔭i,