On first layers of Z p -extensions

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1 Journal of Number Theory ) Contents lists available at SciVerse ScienceDirect Journal of Number Theory On first layers of Z p -extensions Soogil Seo Department of Mathematics, Yonsei University, 34 Sinchon-Dong, Seodaemun-Gu, Seoul , South Korea article info abstract Article history: Received 25 January 202 Revised 6 May 203 Accepted 20 May 203 Available online xxxx Communicated by D. Burns MSC: R27 R29 For any number field k, we show that the elements which generate the first layers of arbitrary Z p -extensions can be characterized by means of a certain norm compatibility property that is defined with respect to the cyclotomic Z p -extension of k. 203 Elsevier Inc. All rights reserved. Keywords: Z p -extensions Universal norm Norm compatible Circular units Kummer theory. Introduction Let Z p be the ring of p-adic integers of the field of p-adic numbers. An extension field K of a number field k will be called a Z p -extension if K/k is a Galois extension and This work was supported by the National Research Foundation of Korea NRF) grant funded by the Korea government MEST) ) and the Basic Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology under Grant address: sgseo@yonsei.ac.kr X/$ see front matter 203 Elsevier Inc. All rights reserved.

2 S. Seo / Journal of Number Theory ) GalK/k) is topologically isomorphic to the additive group of) Z p.inaddition,ifk /k is any Galois extension for which GalK /k) is isomorphic to a non-trivial quotient of Z p, then by the first layer of K /k we shall mean the unique subfield of K which has degree p over k. In this article we shall consider certain subfields of Z p -extensions of k. Weshow,in particular, that if k contains a primitive p-th root of unity, then the elements which generate the first layers of arbitrary Z p -extensions can be characterized in terms of a certain explicit norm compatibility property that is defined in terms of the cyclotomic Z p -extension of k. This characterization then extends in the obvious way to a general number field k: ifζ p denotes a primitive p-th root of unity in a fixed algebraic closure of k), then one can describe the first layers k of Z p -extensions of k by studying the corresponding first layers k ζ p )overkζ p ). The starting point for this article is a result of Bertrandias and Payan in [] which characterizes, for each natural number n, the first layers of Z/p n Z-extensions of a number field k in terms of certain norm groups over the cyclotomic Z/p n Z-extension of k. Our main result is in effect a natural generalization of this result of Bertrandias and Payan. Further, whilst our viewpoint is slightly different from that of [], our methods are somewhat similar. More precisely, since Z p is the inverse limit of the groups Z/p n Z,we shall relate generators of the first layers of Z p -extensions of k to certain inverse limits of generators of subfields of Z/p n Z-extensions for all n. This point of view seems natural since Kummer theory explicitly relates the structures of Galois groups with properties of generators of the relevant extension fields. We now fix a number field k and an algebraic closure k alg of k and for each natural number w we fix a primitive w-th root of unity ζ w in k alg :weassume,aswemay,that ζw l = ζ wl for each divisor l of w. Wealsowriteμ w for the group ζ w of all w-th roots of unity in k alg. We write Qμ p ) for the union of the fields Qμ p n)forn. We also let Q denote the Z p -extension of Q obtained as the fixed field of Qμ p ) under the finite) torsion subgroup of the Galois group GQμ p )/Q). For any number field E we write E cyc for the cyclotomic Z p -extension EQ of E and, for each natural number n, wewritee n for the unique subfield of E cyc which has degree p n over E. Returning to our fixed field k we define a non-negative integer m = m k by the condition that p m is the maximal power of p that divides the order of the torsion subgroup of k or equivalently, ζ p m k and ζ p m+ / k). We assume for the moment that m>0seeremark.3 for the case m =0).In this case we can fix a topological generator τ of Gk cyc /k) = Z p with the property that τζ p n)=ζ +pm p for each integer n m. In this case we also fix an indeterminate T n and can then and indeed will throughout this article) use the correspondence T τ to identify the ring Z[T, + T ) ] with the integral group ring Z[Gk cyc /k)] of Gk cyc /k).

3 402 S. Seo / Journal of Number Theory ) In the sequel we shall consider the following natural projection homomorphisms and π k, : lim k n / Tk n pn k pm) k /k p n an mod Tk n pn k pm)) n N a )modk p π k,2 : lim k n / T p m) k n k pm ) k /k p n an mod T p m) k n k pm )) n N a )modk p. Here each element a n belongs to k n, both inverse limits are defined with respect to the maps induced by the) natural field-theoretic norms k n k n and for each natural number m we write N m for the field-theoretic norm k m k. For the inverse limit lim n k n with respect to the field-theoretic norms, let π k : lim k n k n a n ) n N a ) be the natural projection homomorphism defined similarly as above. Let k comp := Imπ k ) be the elements of k that are norm comparable for the extension k cyc /k. Then it is clear that both of the groups Imπ k, ) and Imπ k,2 ) contain the image in k /k p of) the group k comp. Finally we define a subgroup of k which contains k p ) by setting Θ k := { α k k α /p) is a subfield of a Z p -extension of k }. We are now ready to state our main result. This result shows that if m k > 0, then) the groups Imπ k, ) and Imπ k,2 ) give information about those elements of k whose p-th roots generate first layers of Z p -extensions. Theorem.. Let k be a number field with m k > 0 and for each natural number n let k n denote the n-th layer of the cyclotomic Z p -extension of k. Then one has Imπ k, ) Θ k /k p Imπ k,2 ).

4 S. Seo / Journal of Number Theory ) Remark.2. There are natural examples in which the first inclusion Imπ k, ) Θ k /k p in Theorem. is an equality see Proposition 4.). By using Lemma 3. below it is also possible to prove directly the inclusion Imπ k, ) Imπ k,2 ) of lower and upper bounds in Theorem.. In fact, at this stage, we do not know of any examples in which this inclusion, or even the inclusion k comp k p )/k p Imπ k,2 ), is strict. In particular, for any field k for which k comp k p )/k p =Imπ k,2 ), Theorem. implies that k comp k p )/k p = Θ k /k p and we can then use Coleman s map to describe the generators of all first layers of Z p -extensions of k by using a diagonal embedding from the global p-units into the product of the local completions of k at all p-adic places cf. [6, Theorem 4.] and [2, Theorem 6 and Corollary 7]). Remark.3. If k is any number field for which m k = 0 or equivalently, ζ p / k), then one obviously has Θ k = k p. In this case, however, one can still use Theorem. to give information about the first layers of Z p -extensions of k in the following way. Let S k denote the set of all first layers of Z p -extensions of k. WesetK := kζ p )and foreachfieldk in S k consider the diagram: k K = Kα /p ) k K In this diagram the fields k and K are linearly disjoint over k since [K : k] is prime to p) and so we can recover k asthefixedfieldofkα /p ) under the action of GK/k). In particular, if we write Θ K/k for the subset of Θ K comprising elements α for which there exists a Z p -extension Kα of K which contains Kα /p ) and is Galois over k, then one has S k = { K α /p) GK/k) α Θ K/k }. Finally we note that m K > 0sothatTheorem. applies to the field K to give information about the group Θ K. 2. The work of Bertrandias and Payan We first recall some relevant results from []. To do this we start with the following definition. Definition 2.. For any profinite) group H an extension field K of k is said to be H-extendable over k) if there is an extension field F of K which is Galois over k and such that the Galois group GF/k) is isomorphic to H.

5 404 S. Seo / Journal of Number Theory ) In particular, if K/k is a Galois extension, then K is H-extendable over k if and only if there is a Galois extension F of k which contains K and is such that there is a canonical) short exact sequence of Galois groups GF/K) GF/k) = H GK/k). We write Θ k for the set of all elements α in k such that kα /p )isz p -extendable. Then Θ k obviously contains k p and if k contains a primitive p-th root of unity ζ p,the quotient group Θ k /k ) p is a finite dimensional Z/pZ-vector space. The dimension s k of this space is equal to the maximum number of linearly independent Z p -extensions of k and so s k = r 2 k)++δ k, wherewewriter 2 k) for the number of complex embeddings of k and δ k for the difference between the Z-rank of the group of global units U k of k and the Z p -rank of the image of the p-completion of U k in the product of the local completions of k at all p-adic places. Now Leopoldt s Conjecture is equivalent to asserting that δ k = 0 cf. 5.5 and 3. of [7]). We recall that if k is an abelian field, then this conjecture has been proved by Brumer, and hence for such fields we have dim Z/pZ Θk /k p) = r 2 k)+. ) In the paper [] Bertrandias and Payan also study the larger subgroup Ψ k of k comprising elements α for which kα /p )isz/p n Z-extendable for all natural numbers n. To describe one of their key results we write k cyc for the cyclotomic Z p -extension of k = kμ p m), with m > 0, and for each non-negative integer n we write k n for the unique subfield kμ p n+m) ofk cyc which has degree p n over k. WealsowriteN n for the field-theoretic norm map k n k. Theorem 2.2. See [, Theorem ].) An element α of k is such that kα /p ) is Z/p n Z-extendable if and only if it belongs to the subgroup N n k n )k p. Corollary 2.3. See [, Corollary].) One has Ψ k = n 0 Nn k n ) k p. An important consequence of this result is that there exist extensions which are Z/p n Z-extendable for all n without being Z p -extendable. Indeed, as described in [, 2.4], this follows from the fact that Serre has constructed examples of abelian number fields k for which dim Z/pZ Ψ k /k p ) >r 2 k)+andso) implies that Ψ k is strictly bigger than Θ k.

6 S. Seo / Journal of Number Theory ) The proof of Theorem. In this section we prove Theorem.. We continue to use the notation and hypotheses introduced in Section. In particular, we assume throughout that k contains a non-trivial p-th root of unity so that the integer m = m k ) is strictly positive. We will also use the following convenient abbreviations. For s n, wewriten s,n for the norm map k s k n and N s for the norm map k s k. 3.. The general strategy Our proof of Theorem. is rather involved and so, for the reader s convenience, we will first shortly explain the general strategy. To prove the first inclusion in Theorem. we shall take an arbitrary element of Imπ k, ) and use it to construct, for each natural number n, az/p n Z-extension L n of kμ p n) such that both L n L n+ and L n /k is abelian. Writing L for the union of the fields L n for n we will show that the Z p -rank of GL /k) isatmosttwoandthen that L contains a suitable Z p -extension of k. To prove the second inclusion we will assume to be given an element α of Θ k and fix a Z p -extension F of k which contains kα /p ). Then the composite field F k cyc has Z p -rank at most two over k and we can use Kummer theory to construct a particular sequence of elements ξ n ) n,withξ n kμ p n)foreachn. We are then able to show that the sequence ξ n ) n gives rise to an element of the inverse limit lim n k n / T p m )k n k pm ) and that the image under π k,2 of this element is equal to the image of α in k /k p, as required The first inclusion We now prove the first inclusion Imπ k, ) Θ k /k p that occurs in Theorem.. To do this we fix an element α in Imπ k, ) and then a representative sequence for α of the form ξ n) n with each ξ n in k n and such that N n ξ n ) α mod k ) p and N n,n ξ n = ξ n Tβ pn n a pm n for some β n k n and a n k. In the sequel we shall make much use of the following technical result which is due to Bertrandias and Payan). Lemma 3.. For each pair of natural numbers n and m with n>mthere exists an integer u n m and a polynomial f n m x) in Z[x] such in Z[Gk n m /k)] one has N n m = T p m) f n m T )+ +u n m p m) p n m.

7 406 S. Seo / Journal of Number Theory ) Proof. From [, Lemma ] there exists an integer u n m and polynomial f n mx), for which there is an identity in Z[x] of the form +x + + x pn m = x p m) f n mx)+ +u n m p m) p n m. If we set f n m x) :=f n mx + ), then the claimed equality is obtained by simply substituting the element + T into this identity in place of x. Before proceeding it is convenient to introduce the following abbreviations for fieldtheoretic norm maps for i {0, }. By setting we can find an element ρ i of k such that N,0 := N n m+,n m and N i := N n m+i ξ i := ξ n m+i, N i ξ i = αρ p i. We then define elements ν i and α i by setting and finally define a field ν i := f n m+i T )ξ pm i ρ i and α i := αν p i L /p n+i := k n m+i α n+i ) i. In the next result we record the key properties of these fields. Proposition 3.2. For each natural number n>mand each i {0, } the fields L n+i have all of the following properties. i) L n+i is an abelian extension of k; ii) L n+i is a cyclic extension of k n m+i of degree p n+i ; iii) L n L n+i. Proof. The proofs of claims i) and ii) are actually contained within the proof of [, Theorem b)] but, for the reader s convenience, we shall shortly recall the argument. From Lemma 3., we have α = ρ p N ξ = ρ p +u ξ n m+ p m ) p n m+ fn m+ T ) T p m) ξ

8 S. Seo / Journal of Number Theory ) and hence α pm = ξ +u n m+p m ) p n+ T p m ) f n m+ T )ξ pm ρ p ) = ξ +u n m+p m ) p n+ T p m ) ν p. It follows that T p m ) α = α pm T p m) ν p = ξ +u n m+p m Now let τ be any extension of τ to L n+. Thensince τ α /p n+ )) p n+ = τα )=α +pm x pn+ ) p n+ k p n+ n m+. for some x k n m+ one has τα /pn+ ) L n+ and so L n+/k is a Galois extension. In addition, the group GL n+/k) is abelian if and only if one has τσ α /pn+ ) /p / τ α n+ ) /p = σ τ α n+ ) /p / τ α n+ ). Since τσα /pn+ )/α /pn+ )=σα /pn+ )/α /pn+ ) +pm this condition is in turn equivalent to the condition that α /pn+ ) +pm / τα /pn+ ) is contained in k n m+ which is the fixed field by σ. This condition is satisfied since T p m )α was shown to be a p n+ -th power of k n m+. This shows that L n+ = k n m+ α ) /pn+ ) is an abelian extension of k and cyclic extension of degree p n+ over k n m+, thus proving claims i) and ii). We now turn to the proof of claim iii). To do this we use the compatibility properties of the element ξ n to choose elements β 0 k n m and a 0 k for which one has N,0 ξ = ξ 0 T ) β pn m 0 a p m 0. Then, in order to prove the claim, it is enough to show that Further, since ν = ν 0 = ν ν 0 mod k p n n m+). 2) f n m+ T )ξ pm ρ 0 ρ f n m T )N,0 ξ Tβ pn m 0 a pm 0 ) pm f n m+ T )ξ pm ρ 0 ρ f n m T ) p i= + T )ipn m ξ Tβ pn m 0 a pm 0 ) pm it is enough for us to show that each of the individual factors )

9 408 S. Seo / Journal of Number Theory ) ) p f n m+ T )ξ p i= + T )ipn m f n m T )ξ m, Tf n m T )β pn 0, f n m T )a p2m 0, ρ 0 ρ are p n -th powers of elements in k n m+. Concerning the first factor it can be shown as in [, p. 525] that ) p f n m+ T )ξ p i= + T )ipn m f n m T )ξ = m p i= + T )ipn m ) + u n m p m ) p + u n m+ p m ) p n ) T p m ξ. It is also clear that Tf n m T )β pn 0 belongs to k pn n m+. Further, since T annihilates a 0, Lemma 3. implies that f n m T )a p2m 0 is equal to f n m T ) T p m) a pm 0 = N 0 +u n m p m) p n m) a pm 0 = a u n mp n+m 0 and hence that f n m T )a p2m 0 belongs to k pn n m+. Finally, by applying the norm N 0 to the identity N,0 ξ = ξ 0 Tβ pn m 0 a pm 0, we find that In particular, since N i ξ i = αρ p i N ξ = N 0 ξ 0 N 0 Tβ p ) n m 0 a p n 0 = N 0 ξ 0 a pn 0. for i {0, }, this equality implies that ρ 0 ρ )p = a pn 0 and hence that the factor ρ 0 ρ belongs to k pn n m+. This completes the proofs of the required congruence 2). Returning to the proof that α belongs to Θ k /k p we note that if kα /p )=k 2 then α obviously belongs to Θ k /k p. We shall therefore assume in the sequel that kα /p ) k 2 and hence that the fields kα /p )andk 2 are linearly disjoint over k. In this case the union L of the fields L n for n>mis a Z p -extension of k and L /k is an abelian Galois extension with group GL /k) isomorphic to Z p Z p. It can then be shown that kα /p ) is contained in a Z p -extension of k and hence that α belongs to Θ k /k p ) by following the same argument as in [, Lemma 2]. More precisely, there exists a direct product decomposition of Galois groups of the form G k αν p n ) /p n) /k ) = σ τ where σ generates the group Gk αν p n) /pn )/k )andsokα /p ) is the fixed field of a subgroup H of σ τ of index p. Now any such subgroup H must be of the form σ p,τσ i for some integer i with 0 i<p. In particular, therefore, kα /p )iscontained

10 S. Seo / Journal of Number Theory ) in the fixed field L n of τσ i acting on k αν p n) /pn ). Since the group GL n /k) isisomorphic to Gk αν p n) /pn )/k)/ τσ i = Z/p n Z and L n L n+ by Proposition 3.2iii)) the union of the fields L n for n>mis a Z p -extension of k which contains kα /p ). This shows that α belongs to Θ k /k p, as required The second inclusion We now prove the second inclusion Θ k /k p Imπ k,2 ) that occurs in Theorem.. To do this we fix an element α of Θ k /k p so that kα /p ) is a first layer of a Z p -extension L of k. For each n we write L n for the unique subextension of L which has degree p n over k and choose an element α n m of k n m such that L n k n m = k n m α /pn n m ). Now if the fields L n and k n m are not linearly disjoint over k for any n>m,then one has L = kα /p )=k = kζ /p p ) and hence ζ m p m α mod k ) p.inthiscase we therefore set ξ n := ζ p n+m. Then the sequence ξ n ) n is contained in lim n k n / T p m )k n ) k pm and is also such that N n ξ n )=ξ 0 = ζ p m α mod k ) p so that α = π k,2 ξ n ) n ), as required. In the sequel we may therefore assume that the fields L n and k n m are linearly disjoint over k for all n>mand we shall consider the following diagram of fields. k n m+ α /pn+ n m+ ) k n m+ α /pn n m ) L k n m α /pn n+ k n m+ α /p ) n m ) L n k n m α /p ) L = kα /p ) k n m k = k 0 k n m+ Define ν i and ξ i in k i as follows. Since α i α mod k i )p and α i α i mod k i )pi+m, we write α i = αν p i and α i = α pi+m i β i for some ν i,β i k i.sincek iα /pi+m i )/k for some is abelian, it follows that T p m )α i k i )pi+m, i.e., T p m )α i = ξ pi+m i ξ i k i.noticethatξ i and β i are defined mod μ p i+m and μ pi+m respectively. For i =0,, we use the following notational conventions, ξ i := ξ n m+i, β i := β n m+i, α i := α n m+i, ν i := ν n m+i. It follows from the definitions of ξ i and β i that

11 4020 S. Seo / Journal of Number Theory ) ξ pn+ = ξ pn 0 T p m) β pn. Applying the norm map N,0 to the equation above, we have N,0 ξ p ) n+ = ξ p n+ 0 N,0 T p m ) ) β pn. 3) Applying the absolute norm N 0 and using N 0 T =0,wehave N 0 N,0 ξ p )) n+ = N0 ξ p ) n+ 0 N,0 β pn+m. By taking a p n+ -th root in the above equation, we have N 0 N,0 ξ ) ) ) N 0 ξ0 N,0 β pm mod μp n+ k = μ p m = N 0 ξ0 N,0 β pm mod μ p n). It follows from the surjection of the norm map over the roots of unity that there is δ 0 k n m andanelementζ of μ p n such that N,0 ξ )/ξ 0 = N,0 β pm Tδ 0 ζ. 4) By raising this equality to the power p n+ and then substituting the equality 3) into the resulting expression we find that This leads to It follows that N,0 β pn+m Tδ pn+ 0 = T p m) N,0 β pn = TN,0 β pn /N,0β pm+n. TN,0 δ p n 0 ) = N,0 Tδ p n 0 ) = Tδ p n+ 0 = TN,0 β pn. T N,0 δ 0 β ) ) = ν = Tν 2 for some ν μ p n, ν 2 μ p n+m. Hence, we have N,0 δ 0 β ) ν 2 k. This shows that ν 2 μ p n, i.e., Now Eq. 4) implies that N,0 δ 0 β ) k μ p n. 5) N,0 ξ )/ξ 0 N,0 β pm Tδ 0 N,0 δ 0 β ) pm T p m) δ 0 mod μ p n), and this combines with 5) to imply that ξ n) n 0 defines an element of the inverse limit lim k n /μ p n+mt p m )k n k pm.

12 S. Seo / Journal of Number Theory ) Since the norm maps are stable over p n -th roots of unity as n varies this fact in turn combines with Lemma 3.3 below to show that we may use the ambiguity of an element of μ p n+m that is involved in the definition of the elements ξ n to ensure that) we can assume ξ n ) n lim k n / T p m) k n k pm. Finally, we will show that N ξ ) α mod k p μ p m. It follows from α = αν p and T p m )α = ξ pm+ that ξ pm+ = T p m) α = α pm T p m) ν p and hence by applying the norm operator N )that α pm+ = N ν pm+ N ξ pm+ and T p m )ν ξ pm α pm mod μ p. Applying Lemma 3., wehave N ν = T p m) f T )ν ν +u p m )p f T ) ξ pm α pm ) ν +u p m )p mod μ p. It follows from f T )α pm = T p m )f T )α = N + u p m )p)α = α u p m+ that N ν k p k = k p μ p m. We let N ν c p mod μ p m for some c k. This leads to α pm+ = N ν pm+ N ξ pm+ = c pm+2 N ξ pm+ and hence, α c p N ξ mod μ p m as claimed. Since a p m -th root of unity gives rise to a norm comparable sequence over k cyc /k we can construct an inverse limit ξ n) n such that α N ξ mod k p. This completes the proof of Theorem.. The following result is well-known and straightforward to prove). Lemma 3.3. Let A n,f m,n ), B n,g m,n ) and C n,h m,n ) be inverse systems. If for each m there exists a short exact sequence ρ 0 A m B m m C m 0

13 4022 S. Seo / Journal of Number Theory ) which is compatible with the transition morphisms f m,n, g m,n and h m,n for all m n, then there exists an associated exact sequence lim 0 lim A n lim B ρ n n lim C n Coklim ρ n ) 0. Further, if for each n the image of f m,n is stable for all sufficiently large m, then the group Coklim ρ n ) vanishes. 4. Abelian fields In this section we assume that k is an absolutely abelian field and explain how in this case the group of circular units plays a particularly important role in the context of Theorem.. We start with a result which shows, in particular, that the first inclusion in Theorem. is in some important cases an equality. Proposition 4.. Let k denote the p m -th cyclotomic field Qμ p m) for some m>0. Then if k validates the Kummer Vandiver conjecture one has equalities Imπ k, )=Θ k /k p = C k k p) /k p where C k is the group ζ pm)z[gk/q)] of circular p-units of k. Proof. For each natural number n we shall respectively write C n, C n, U n and U n for the groups of circular p-units, circular units, p-units and units of the field k n.forthe definitions of the groups C n and C n and details of the class number formula of Sinnott that we now use we refer to [4] for the cyclotomic case and [5] for the general abelian case.) Assuming that k validates the Kummer Vandiver conjecture at p, the class number formula of Sinnott implies that for each n 0 the groups U n/c n Z p and U n /C n Z p both vanish. In particular, writing U k for the group of p-units of k the norm compatibility property of the circular p-units ζ p m+n for n 0 implies that U k k p) /k p = C k k p) /k p Imπ k, ) Θ k /k p. 6) On the other hand, one has dim Z/pZ U k k p) /k p) =dim Z/pZ U k / U k ) p ) = r2 k)+=dim Z/pZ Θk /k p) where the first equality follows from U k k p =U k )p, the second from the fact that U k contains ζ p and has rank r 2 k), and the last from ). This shows that the inclusions in 6) must be equalities, as claimed.

14 S. Seo / Journal of Number Theory ) Remark 4.2. If one tries to prove the result of Proposition 4. for an arbitrary abelian field k rather than a cyclotomic field), then the necessary computations become considerably more involved. Nevertheless, it seems likely to us that, at least under the assumed validity of the Greenberg conjecture from [3]), one could describe explicitly the connection between Imπ k, ) and the group of circular p-units of k. However, for the moment we prefer to omit any further details of this more complicated case. Finally we note that for a general abelian number field k, one can combine Theorem. with the basic properties of circular units to explicitly describe groups of elements whose p m -th roots generate first layers of certain Z p -extension. To explain this we let k = kμ p m) be an abelian number field of conductor p m s with p s. Forr>0, write F r = Qμ r ). For a Galois extension K/k, welettr K/k = σ GK/k) σ. Then, one can check easily that for any divisor s of s and any element a n,s ) n of the inverse limit lim n Z[GF pn s /Q)], the sum s s Tr Fp n s /k s a n,s ζp nζ p n s ) forms a norm compatible system inside k /k as n varies, where k s = k F p n s and the inverse limit is taken with respect to the natural restriction maps. In particular, since the natural projection map lim Z [ GF pn s /Q)] Z [ GF pm s /Q)] n m is surjective, Theorem. implies that a p m -th root of any element of the Galois module generated by the elements Tr Fp n s /k ζ s p mζp m s ) generates a field that is contained in a Z p -extension of k. Acknowledgment We would like to thank the referee for his helpful comments and suggestions. References [] F. Bertrandias, J. Payan, Γ -extensions et invariants cyclotomiques, Ann. Sci. Ecole Norm. Sup ) [2] R. Coleman, Division values in local fields, Invent. Math ) [3] R. Greenberg, On the Iwasawa invariants of totally real number fields, Amer. J. Math ) [4] W. Sinnott, On the Stickelberger ideal and the circular units of a cyclotomic field, Ann. of Math ) [5] W. Sinnott, On the Stickelberger ideal and the circular units of a cyclotomic field, Invent. Math ) [6] T. Tsuji, Semi-local units modulo cyclotomic units, J. Number Theory ) 26. [7] L.C. Washington, Introduction to Cyclotomic Fields, second edition, Grad. Texts in Math., vol. 83, Springer-Verlag, 997.