A criterion for Lubin’s conjecture

– We prove that a formulation of a conjecture of Lubin regarding two power series commuting for the composition is equivalent to a criterion of checking that some extensions generated by the nonarchimedean dynamical system arising from the power series are Galois. As a consequence of this criterion, we obtain a proof of Lubin’s conjecture in a new case


Introduction
Let K be a finite extension of Q p , with ring of integers O K and maximal ideal m K .

Families of power series in T • O K [[T ]] that commute under composition have been studied
by Lubin [Lub94] under the name of nonarchimedean dynamical systems, because of their interpretation as analytic transformations of the p-adic open unit disk.This study series to commute with a noninvertible series, there must be a formal group somehow in the background".
In this note, we consider two power series P, U ∈ T • O K [[T ]] such that P • U = U • P , with P ′ (0) ∈ m K and U ′ (0) ∈ O × K , and we assume that P (T ) = 0 mod m K and that U ′ (0) is not a root of unity.Our so called version of Lubin's conjecture is the following: K not a root of unity, and such that P (T ) = 0 mod m K .Then there exists a finite extension E of K, a formal group S defined over O E , endomorphisms of this formal group P S and U S and a power series h(T In the conjecture above, we say following Li's terminology [Li97a] that P and P S are semiconjugate and that h is an isogeny from P S to P .
In several proven cases of this conjecture [Sar05, Ber17, Spe18], the Lubin-Tate formal group is actually defined over O K .However, this is not true in general.
The goal of this note is to prove the following theorem, which gives a new criterion to prove Lubin's conjecture in some cases: Theorem 0.2.-Let (P, U) be a couple of power series in T • O K [[T ]] such that P • U = U • P , with P ′ (0) ∈ m K and U ′ (0) ∈ O × K , and we assume that P (T ) = 0 mod m K and that U ′ (0) is not a root of unity.Then there exists a finite extension E of K, a Lubin-Tate formal group S defined over O L , where E/L is a finite extension, endomorphisms of this formal group P S and U S over O E , and a power series h(T 2. there exists a finite extension E of K and a sequence (α n ) n∈N where α 0 = 0 is a root of Q and Q(α n+1 ) = α n such that for all n ≥ 1, the extension E(α n )/E is Galois.
The proof relies mainly on the same tools and strategy used in [Poy22], which are the tools developed by Lubin in [Lub94] to study p-adic dynamical systems, the "canonical Cohen ring for norms fields" of Cais and Davis [CD15] and tools of p-adic Hodge theory following Berger's strategy in [Ber16a].
As a corollary of our main theorem, we obtain the following result, which is a new instance of Lubin's conjecture: In order to prove our main theorem, we also need to prove that some extensions are strictly APF, which is a technical condition on the ramification of the extension.Cais and Davis have considered in [CD15] what they called "ϕ-iterate" extensions, and later on proved with Lubin that those extensions are strictly APF [CDL16].Here, we show that that this result still holds for more general extensions which generalize the ϕ-iterate extensions of Cais and Davis: Theorem 0.4.-Let K ∞ /K be an extension generated by a sequence (u n ) of elements of Q p such that there exists a power series where d is a power of the cardinal of k K , and an element π 0 of m K such that u 0 = π 0 and Organization of the note.-The first section recalls the construction and properties of some rings of periods which are used in the rest of the paper.The second section is devoted to the proof of theorem 0.4, using the rings of periods of the first section in order to do so.In the third section we recall the main result of [Lub94] which explains why "Lubin's conjecture" seems reasonable.In section 4, we prove that our version of Lubin's conjecture implies that the two conditions of theorem 0.2 are satisfied.Section 5 and 6 show how to use p-adic Hodge theory, using the same strategy as in [Poy22], along with results from [Lub94], in order to prove that the infinite extension generated by such a Q-consistent sequence is actually generated by the torsion points of a formal Lubin-Tate group.In section 7, we show how to use the "canonical Cohen ring for norms fields" of Cais and Davis [CD15] to prove that there is indeed an isogeny from an endomorphism of a formal Lubin-Tate group to Q. Section 8 is devoted to the proof of theorem 0.3.

Rings of periods
Let K be a finite extension of Q p , with uniformizer π K , and let K 0 = Q unr p ∩ K denote the maximal unramified extension of Q p inside K. Let q = p h be the cardinality of k K , the residue field of K, and let e be the ramification index of K, so that eh = [K : F be the set of elements of F such that v K (x) ≥ c.We now recall some definition of properties of some rings of periods which will be used afterwards.We refer mainly to [CC98] [Fon94] for the properties stated here.The slight generalization to the classical rings by tensoring by O K over O K 0 can for example be found in [Ber16b]. Let O Cp /a c Cp .This is the tilt of O Cp and is perfect ring of characteristic p, whose fraction field E is algebraically closed.It is endowed with a valuation v E induced by the one on K.We let

vectors, and let
Any element of A (resp.A + ) can be uniquely written as i≥0 π k K [x i ] with the x i ∈ E (resp.E + ).We let w k : A−→R ∪ {+∞} defined by w k (x) = inf i≤k v E (x i ).
For r ∈ R + , we let A †,r denote the subset of A of elements x such that w k (x) + pr e(p−1) k is ≥ 0 for all k and whose limits when k−→ + ∞ is +∞.We let n(r) be the smallest integer such that r ≤ p nh−1 (p − 1).
We also let , where x 0 = x, and w i (x) + pr e(p−1) i ≥ 0 for all i between 0 and k − 1.Now we can write x [x] ∈ A as i≥0 π i K [y i ], where y i = x i x for i between 0 and k − 1.In particular, y 0 = 1.Now a direct computation leads to the fact that w , we obtain that its inverse also lies into A †,r ′ + π k K A.
Let ϕ q : E + → E + denote the map x → x q .This extends to a map E → E also given by x → x q , and by functioriality of Witt vectors those maps extend into maps ϕ q on A + and A.
Recall that there is a surjective map θ :

Strictly APF extensions
A theorem of Cais, Davis and Lubin [CDL16] gives a necessary and sufficient condition for an infinite algebraic extension L/K to be strictly APF.In particular, this condition implies that what Cais and Davis have called a "ϕ-iterate" extension in [CD15] is strictly APF.
Recall that a (slight generalization of what Cais and Davis in [CD15] have called a) ϕ-iterate extension K ∞ /K is an extension generated by a sequence (u n ) of elements of Q p such that there exists a power series where d is a power of the cardinal of k K , and a uniformizer π 0 of O K such that u 0 = π 0 and The main theorem of [CDL16] gives a necessary and sufficient condition for an infinite algebraic extension L/K to be strictly APF, and in particular implies directly that those ϕ-iterate extensions are strictly APF.
In this section we will prove that this result remain true if we remove the assumption in the definition above that π 0 is a uniformizer of O K , and instead just assume that π 0 ∈ m K .We even allow π 0 to be equal to 0, which is basically what we'll consider when looking at consistent sequences attached to a noninvertible stable power series.
If L is a finite extension of Q p , we let v L denote the p-adic valuation on L normalized such that v L (L × ) = Z, and we still denote by v L its extension to For the rest of this section, we let , where s is a power of the cardinal of k K , we let π 0 be any element of m K , and we define a sequence (v n ) n∈N of elements of Q p as follows: we let v 0 = π 0 , and for n ≥ 0, we let v n+1 be a root of P (T ) − v n .We let K n = K(v n ) the field generated by v n over K, and we let K ∞ = n K n .If v 0 = 0, then we choose v 1 to be = 0, so that the null sequence is excluded from our considerations.
Proposition 2.1.-There exists n 0 ≥ 0 and d ≥ 1 such that, for all n ≥ n 0 , we have v Kn (v n ) = d and the extension K n+1 /K n is totally ramified of degree s.
Proof.-The fact that the Weierstrass degree of P is greater than 1 along with Weierstrass preparation theorem show that the sequence v p (v n ) is strictly decreasing.In particular, there exists n 0 ≥ 0 such that for n ≥ 0, the Newton polygon of P − v n has only one slope, equal to 1 s v p (v n ).This implies that for n ≥ n 0 , we have , so that the sequence (d n ) n∈N is decreasing.Since this sequence takes its values in N, it is stationary and therefore there exists n 1 ≥ n 0 such that, for all n ≥ n 1 , d n+1 = d n .In particular, this implies that the inequalities above are all equalities and thus that for n ≥ 1, s = [K n+1 : K n ] and that K n+1 /K n is totally ramified, and we can take Let us write d = p k m where m is prime to p.
Since P (T ) = T s mod m K , the sequence (v n ) gives rise to an element v of E + = lim ← − x →x s O Cp /π 0 .We let ϕ s denote the s-power Frobenius map on E + and A + .

Proposition 2.2. -There exists a unique
Proof.-One can use the same argument as in [CD15, Rem.7.16] to produce an element in A + such that P (v) = ϕ s (v) and such that θ • ϕ −n s (v) = v n (note that one also needs to extend the results from ibid to the case where the Frobenius is replaced by a power of the Frobenius, which is straightforward).
Such an element automatically lifts v by definition of the theta map.For the uniqueness, one checks that the map x → ϕ −1 s (P (x)) is a contracting map on the set of elements of )) and is thus unique.
Since E is algebraically closed, there exists u ∈ E such that u m = v.Since such a u necessarily has positive valuation, it actually belongs to E + .
Since P (T ) = T s mod π 0 , we can write which is well defined because m is prime to p.Note that Q(T ) is overconvergent, meaning that it converges on some annulus bounded by the p-adic unit circle.
Proof.-We first construct u such that ϕ s (u) = Q(u).Just as the proof as in 2.2, the map ) is a contracting map on the set of elements of A lifting u, so that )) and is unique.Therefore, there exists ]. Now assume that there exists some k ≥ 1 and r ′ > 0 such that u ∈ A †,r ′ + π k 0 .We can thus write u = u k + π k 0 z k , where u k ∈ A †,r ′ and z k ∈ A. We have Recall that since u ∈ A † , there exists some r > 0 such that u ∈ A †,r and there exists n(r) ≥ 0 such that, for all n ≥ n(r), the element u n := θ • ϕ −1 s (u) is well defined and belongs to O Cp .Actually, since u m = v, we have that u m n = v n , and in particular we know that v K (u n )−→0.

Lemma 2.4.
-There exists a constant c > 0, independent of n, such that for any n ≥ n(r) and for any g ∈ G Kn and any i ≥ 1, we have Since m is fixed and v K (u n )−→0, it suffices to prove that there exists c > 0 independent on n such that v K (g(v n+i ) − v n+i ) ≥ c for all g ∈ G Kn .
Since P (T ) = T s mod m K , and since P •j (v n+i ) = v n , we already know that for all n ≥ 0 and for all g ∈ G Kn , we have s is a power of p, and let j ≥ 0 be such that s j ≥ p k > s j−1 .Let f ≥ 0 be such that p −f s j = p k .In particular, we have v Kn (u For n ≥ n 0 , let π n denote a uniformizer of O Kn .Since for all n ≥ n 0 the extensions K n+1 /K n are totally ramified, the minimal polynomial of π n+1 over K n is an Eisenstein polynomial, and we choose the π n so that N K n+1 /Kn (π n+1 ) = π n for all n ≥ n 0 .
Lemma 2.5.-For any n ≥ n(r), we can write and that both elements belong to O K n+j , so that we can write Taking the m-th root, this implies that there exists where the coefficients belong to Theorem 2.6.-The extension K ∞ /K is strictly APF.
Proof.-In order to prove the theorem, it suffices by [Win83, Prop.1.2.3] to prove that the extension • K is strictly APF, it suffices to prove that the v K valuations of the non constant and non leading coefficients of the Eisenstein polynomial of π n+1 over F (m) • K n , for n ≥ n 0 , are bounded below by a positive constant independent of n, so that • K n 0 satisfies the criterion of the main theorem (Thm 1.1) of [CDL16].Let n ≥ n 0 .
By the lemma 2.5 and by induction, we can write where the h i belong to k F (m) .
Let g ∈ G F (m) •Kn .We have where all the terms on the RHS have v K -valuation at least equal to c > 0 by lemma 2.4, The conjugates of π n+1 over K n are the elements g(π n+1 ), for g ∈ G Kn , and satisfy the conditions v K (g(π n+1 ) − π n+1 ) ≥ c > 0, which ensures that the v K valuations of the non constant and non leading coefficients of the Eisenstein polynomial of π n+1 over F (m) • K n are bounded below by a positive constant independent of n, which is what we wanted.

Non archimedean dynamical systems
Let K be a finite extension of Q p , with ring of integers O K , uniformizer π, maximal ideal m K and residual field k of cardinal q = p h .We let K 0 = K ∩ Q nr p be the maximal unramified extension of Q p inside K and we let O K 0 denote its ring of integers.We let C p denote the p-adic completion of In this note, we assume that the situation is "interesting", namely that P (T ) = 0 mod m K and that U ′ (0) is not a root of unity.

Proposition 3.1. -There exists a power series H(T ) ∈ T • k[[T ]] and an integer
Proof.-This is theorem 6.3 and corollary 6.2.1 of [Lub94].
Near the end of his paper [Lub94], Lubin remarked that "Experimental evidence seems to suggest that for an invertible series to commute with a noninvertible series, there must be a formal group somehow in the background."This has led some authors to prove some cases (see for instance [Li96] ]) of this "conjecture" of Lubin.The various results obtained in this direction can be thought of as cases of the following conjecture: K not a root of unity, and such that P (T ) = 0 mod m K .Then there exists a finite extension E of K, a formal group S defined over O E , endomorphisms of this formal group P S and U S , and a power series h(T Remark 3.3.-While in many instances of the cases where this conjecture is proven, the formal group is actually defined over O K [Sar05, Ber17, Spe18], one can produce instances where the formal group is defined over the ring of integers of a finite unramified extension of O K [Ber19,§3].The author does not know of a case where the extension E the formal group is defined over is ramified over K so it might be possible that the assumption that E is an unramified extension of K can be enforced.

Endomorphisms of a formal Lubin-Tate group
not a root of unity, and such that P (T ) = 0 mod m K .In this section, we assume that there exists a finite extension E of K, a Lubin-Tate formal group S defined over O L with ] and an endomorphism P S of S such that h is an isogeny from P S to P .[̟ d L ]) = Card(k L ) d .We let V denote the power series commuting with P such that h Let (u n ) n∈N be a sequence of elements of Q p such that u 0 = 0 is a root of Q S , and Q S (u n+1 ) = u n .In Lubin's terminology (see the definition on page 329 of [Lub94]), the Then for all n ≥ 1, the extensions E n /E are Galois.
Let Q as in lemma 4.1 and let is Q-consistent follows directly from the fact that h is an isogeny from Q S to Q.

Embeddings into rings of periods
Let L := K n 0 with n 0 as in proposition 2.1.Since P (T ) = T p d mod m K , there exists m ≥ 1 such that P •m acts trivially on k L , so that the degree r of Q is a power of the cardinal of k L , and we let Q := P •m after having chosen such an m.We let w 0 = v n 0 and x [F ′ :F ] − 1) ≥ c.We can always assume that c ≤ v L (p)/(p − 1) and we do so in what follows.By §2.1 and §4.2 of [Win83], there is a canonical G L -equivariant embedding ι L : A L (K ∞ ) ֒→ E + , where A L (K ∞ ) is the ring of integers of X L (K ∞ ), the field of norms of K ∞ /L.We can extend this embedding into a G L -equivariant embedding X L (K ∞ ) ֒→ E, and we note E K its image.
It will also be convenient to have the following interpretation for E + : To see that this definition coincides with the one given in §1, we refer to [BC09, Prop.

4.3.1].
Note that, even though E K depends on K ∞ rather than on L, it is still sensitive to L: The sequence (w n ) defines an element w ∈ E + .Proposition 5.2.-There exists a unique w ∈ A + lifting w such that Q(w) = ϕ r (w).Moreover, we have that θ Proof.-This is the same proof as for the proposition 2.2.
For all k ≥ 0, we let Proof.-Note that for all k ≥ 0, R k is an O L -algebra, separated and complete for the π L -aidc topology, where We know by the theory of field of norms that lim Since the valuation on R/π L R is discrete, and since this set is nonempty because it contains the image of the element w given by proposition 5.2, such an element u exists, and we have and complete for the π L -adic topology.
Proposition 5.4.-There exists k 0 ≥ 0 such that, for all k ≥ k 0 , we can take z k+1 = ϕ −1 r (z k ) and we let z = z k 0 .
Proof.-The proof of proposition 5.
) by construction of the z k .This implies that the sequence (v(k)) k≥n 0 is nonincreasing, and since it is bounded below by 1, this implies that there exists some k 0 ≥ n 0 such that, for all k ≥ k 0 , we have and by construction of the z k this implies that we can take z k+1 = ϕ −1 r (z k ) which concludes the proof.
We now let k 0 be as in proposition 5.4.Note that in particular, for all k ≥ k 0 , we have ) − a and we let S(T ) = R(T ) − a so that ϕ r (z ′ ) = S(z ′ ).For S(0) to be 0, it suffices to find a ∈ m L such that R(a) = a.Such an a exists since we have R(T ) ≡ T r mod m L so that the Newton polygon of R(T ) − T starts with a segment of length 1 and of slope −v p (R(0)).Now, we have S(z ′ ) = ϕ r (z ′ ) and so S(z ′ ) = z ′ r , so that S(T ) ≡ T r mod m L .
Lemma 5.5 shows that one can choose z ∈ ], and we will assume in what follows that such a choice has been made.Lemma 5.6.-Assume that there exists m 0 ≥ 0 such that for all m ≥ m 0 , the extension L m /L m 0 is Galois.Then the ring O L [[z]] is stable under the action of Gal(K ∞ /L m 0 ), and if g ∈ Gal(K ∞ /L m 0 ), there exists a power series is stable under the action of Gal(K ∞ /L m 0 ), and by proposition 5.4, this set is equal to , and thus H g (z) = g(z).

p-adic Hodge theory
Let us assume that there exists m 0 ≥ 0 such that for all m ≥ m 0 , the extension L m /L m 0 is Galois.Lemma 5.6 shows that in this case we are in the exact same spot as the situation after lemma 5.15 of [Poy22].In particular, the exact same techniques apply.
We keep the notations from §4 and we let κ L is injective and crystalline with nonnegative weights.
Proof.-This is the same as corollary 5.17 and proposition 5.19 of [Poy22].
For λ a uniformizer of L m 0 , let (L m 0 ) λ be the extension of L m 0 attached to λ by local class field theory.This extension is generated by the torsion points of a Lubin-Tate formal group defined over L m 0 and attached to λ, and we write χ the corresponding Lubin-Tate character.Since K ∞ /L m 0 is abelian and totally ramified, there exists λ a uniformizer of Proof.-Theorem 5.27 of [Poy22] shows that there exists F ⊂ L m 0 and r ≥ 1 such that κ = N Lm 0 /F (χ Lm 0 λ ) r .The fact that κ takes its values in O × L shows that F is actually a subfield of L.
Recall that relative Lubin-Tate groups are a generalization of usual formal Lubin-Tate group given by de Shalit in [dS85].Theorem 6.3.-There exists F ⊂ L and r ≥ 1 such that κ = N L/F (χ L λ ) r .Moreover, there exists a relative Lubin-Tate group S, relative to the extension Theorem 7.4.-Let (P, U) be a couple of power series in T K , and we assume that P (T ) = 0 mod m K and that U ′ (0) is not a root of unity.Then there exists a finite extension E of K, a Lubin If those two conditions are satisfied, then proposition 7.3 shows that there exists a finite extension E of K, a subfield F of E, a relative Lubin-Tate group S, relative to the extension F unr ∩ E of F , and an endomorphism Q S of S such that there exists an isogeny from Q S to Q. Thus there exists an isogeny from an endomorphism P S of S to P .In order to conclude, it suffices to notice that a relative Lubin-Tate formal group S, relative to an extension F unr ∩ E of F is actually isomorphic over F unr ∩ E to a Lubin-Tate formal group S ′ defined over F .

A particular case of Lubin's conjecture
We now apply the results from the previous sections to the particular case where ] such that P • U = U • P , with P (T ) = T p mod m K and U ′ (0) ∈ O × K not a root of unity.We consider as in §3 a P -consistent sequence (v n ) and we let K n = K(v n ) for n ≥ 0. We let n 0 be as in proposition 2.1.Proposition 8.1.-There exists m 0 ≥ 0 such that for all m ≥ m 0 , the extension K m /K m 0 is Galois.
Proof.-By [Lub94, Prop.3.2], the roots of iterates of P are exactly the fixed points of the iterates of U. Up to replacing U by some power of U, we can assume that U ′ (0) = 1 mod m K and that there exists n ≥ n 0 such that U(v n ) = v n but U(v n+1 ) = v n+1 (since U(T ) − T admits only a finite number of roots in the unit disk).
Since U(v n ) = v n and U commutes with P , this implies that U(v n+1 ) is also a root of P (T ) − v n .The discussion on page 333 of [Lub94] shows that the set {U •k (v n+1 )} k∈N has cardinality a power of p, and is not of cardinal 1 since U(v n+1 ) = v n+1 by assumption.
Since P (T ) − v n has exactly p roots, this implies that the set {U(v n+1 )} has cardinality p, and thus all the roots of P (T ) − v n are contained in K n+1 , so that K n+1 /K n is Galois.
Let m > n.The extension K m /K n is generated by all the roots of P Since U swaps all the roots of P (T ) − v n , it is easy to see that the U-orbit {U •k (v m )} k≥0 contains all the roots of P •(m−n) (T ) − v n , so that K m /K n is Galois.
This prove the proposition.
We are now in the conditions of our theorem 7.4, which yields the following: Corollary 8.2.-Lubin's conjecture is true for (P, U).
and only if the following two conditions are satisfied: 1. there exists V ∈ T • O K [[T ]], commuting with P , and an integer d ≥ 1 such that Q(T ) = T p d mod m K where Q = V • P ;

Lemma 4. 1 .
-There exists V ∈ T • O K [[T ]], commuting with P , and an integer d ≥ 1 such that Q(T ) = T p d mod m K where Q = V • P .Moreover, there exists Q S endomorphism of S such that h is an isogeny from Q S to Q. Proof.-First note that for any V S invertible series commuting with P S , there corresponds an invertible power series V commuting with P .Since S is a formal Lubin-Tate group over O L , P S corresponds to the multiplication by an element α ∈ m L .Let [̟ L ] denote the multiplication by ̟ L on S, a uniformizer of O L such that [̟ L ](T ) = T Card(k L ) mod m L (we can find such a uniformizer since S is a Lubin-Tate formal group defined over O L ).Since α ∈ m L , there exists c ∈ O × L and an integer d ≥ 1 such that α = c • ̟ d L .In particular, we have wideg([α]) = wideg(P) = wideg( Lemma 5.5.-The ring O L[[z]] is stable by ϕ r .Moreover, there exists a ∈ m L such that if z ′ = z − a then there exists S(T) ∈ T • O L [[T ]] such that S(z ′ ) = ϕ r (z ′ ) and S(T ) ≡ T r mod m L .Proof.-The set x ∈ A + , θ • ϕ −n r (x) ∈ O L n+k0 for all n ≥ 1 is clearly stable by ϕ r and equal to O L [[z]] by proposition 5.4, so that ϕ r (z) ∈ O L [[z]] and so there exists R ∈ O L [[T ]] such that R(z) = ϕ r (z).In particular, we have R(z) = z r and so R(T ) ≡ T r mod m L .Now let R(T ) = R(T + a) with a ∈ m L and let z is not a root of unity.Then there exists a finite extension E of K, a Lubin-Tate formal group S defined over O L , where E/L is a finite extension, endomorphisms of this formal group P S and U S over O E , and a power series h and the extensions E(v n )/E are Galois for all n ≥ 1.Proof.-We know that E n /E are Galois abelian extensions.Since E ⊂ E(v n ) ⊂ E n , this implies that the extensions E(v n )/E are Galois.The fact that the sequence (v n ) n∈N n+k is the image of ring of integers of the field of norms of L ∞ /L k inside E by the embedding ι L , and we will denote lim ← −x →x r O L n+k /a c L n+k by Y k .We normalize the valuation of Y k so that v Y k (Y k ) = Z.By proposition 5.1, we get that for k ≥ n 0 , we have Y k+1 = ϕ −1 r (Y k ) and thus the valuation v Y k+1 is equal to rv -Tate formal group S defined over O L , where E/L is a finite extension, endomorphisms of this formal group P S and U S over O E , and a power series h(T ) ∈ T O E [[T ]] such that P • h = h • P S and U • h = h • U S ifand only if the following two conditions are satisfied: 1. there exists V ∈ T • O K [[T ]], commuting with P , and an integer d ≥ 1 such that Q(T ) = T p d mod m K where Q = V • P ; 2. there exists a finite extension E of K and a sequence (α n ) n∈N where α 0 = 0 is a root of Q and Q(α n+1 ) = α n such that for all n ≥ 1, the extension E(α n )/E is Galois.Proof.-Lemmas 4.1 and 4.2 of §2 imply that if such a Lubin-Tate formal group exist then the two conditions are satisfied.