Relative Spectral Norm on Algebraic Numbers

Relative Spectral Norm on Algebraic Numbers

ANGELPOPESCU(*) - SOBIASULTANA(**)

ABSTRACT- LetK be a subfield ofQ, a fixed algebraic closure ofQ, the field of rational numbers. LetGKˆGal(Q=K) be the absolute Galois group ofK. For anyx2Q, we consider theK-spectral norm:kxk_Kˆmaxfjs(x)j:s2GKg. Let ebe the conjugation automorphism ofQand letC(GK) be the Banach algebra of all continuous mappings defined on the compact groupGKwith values inC. Let Csym(GK) be the Banach algebra overKeof all symmetric functions ofC(GK). Here KeˆRorCis the completion ofK relative to the usual absolute valuej:j. A function f is said to be symmetric if f(es)ˆe(f(s)) for any s2GK (when e2GK). LetQeKbe the completion ofQwith respect tok:k_K. In this paper we prove thatQeK Csym(GK) ife2GKandQeK C(GK) ife2=GK. These last iso- morphisms are K-isomorphisms of Banach algebras. We give some othere properties of the closed subalgebras ofC(GK) in connection with some subfields of algebraic numbers.

1. The notion of a spectral norm on the algebraic number field was in- troduced by A. Popescu, N. Popescu and A. Zaharescu in [13]. This is a normk:konQ, a fixed algebraic closure ofQ, the field of rational numbers, defined byx ! kxk ˆmaxfjs(x)j:s2Gal(Q=Q)g. The completion Qe of Qrelative to this norm is a ring with zero divisors but with some interesting arithmetical and topological properties (see [8], [10], [11] and [13] - [17]).

The main property ofQe is that the absolute Galois groupGˆGal(Q=Q) of Qacts continuously on it, which is not the case ofC(only the identityeand the conjugation automorphismeofGact continuously onC). So that, if we

(*) Indirizzo dell'A.: Technical University of Civil Engineering Bucharest, Department of Mathematics, B-ul Lacul Tei 124, 72302 Bucharest 38, Romania.

E-mail: [email protected]

(**) Indirizzo dell'A.: Abdus Salam School of Mathematical Sciences, 68-B, New Muslim Town, Lahore, Pakistan.

E-mail: [email protected]

1991 Mathematics Subject Classification. Primary 11R99, 12J10; Secondary 46J10, 11R32.

want to get information onGby its action on different topological spaces, a good candidate isQ. Such type of researches were also done for the locale absolute Galois group GpˆGal(Qp=Qp), where Qp is a fixed algebraic closure of thep-adic number fieldQ_p(see [4], [9], and [12]). Some ideas of these researches come directly or indirectly from [2], [3].

In a letter to the first author [19], Prof. A. Zaharescu had the idea to generalize the usual spectral normk:kto aK-spectral norm onQ:

x ! kxk_Kˆmaxfjs(x)j:s2G_KˆGal(Q=K)g;

whereKis an arbitrary subfield ofQ.

In this paper we study the completionQe_KofQwith respect to theK- spectral norm defined above. Since the techniques of [13] - [17] are sometimes not useful in this new frame, we introduce some new ones. Let C(GK) be theC-Banach algebra of all continuous mappings defined on the compact groupG_KˆGal(Q=K) (relative to its Krull topology) with values inC(with its usual absolute value topology). An elementf 2 C(G_K) is said to be symmetric (only in the case whene2G_K) iff(es)ˆe(f(s)) for any s2G_K. Ife2= G_Kwe cannot define such a notion. LetC_sym(G_K) be theK- subalgebra of C(GK) which consists of all symmetric mappings ofC(GK).

LetLeKbe the topological closure inQeK of a subfieldLofQ,LK, with respect tok:k_K. In Theorem 1 and Remark 2 we prove that ifKR\Q, then Qe_K is isomorphic and isometric with C_sym(G_K) as K-algebras. Ife K6R\Q( i.e.e2= G_K), thenQe_K is isometric and isomorphic asC-alge- bras with C(GK) (Theorem 2). In Theorem 3 we prove that always QeK[ei] C(GK), whereeiis the constant functionf(s)ˆiˆ 

p 1

for anys inG_K. SoW(Qe_K) has the codimension 1 or 2 inC(G_K). HereW:Q ! C(G_K), is the following map: W(a)ˆW_a, where W_a(s)ˆs(a) and W is the unique extension by continuity of W to Qe_K. In Theorems 4, 5 and 6 we obtain analogous results as in Theorems 1, 2 and 3 for a subextension L=K of Q=K. Theorem 7 is the direct generalization of Theorem 2.2 of [13], namely Le_KˆK[x] for anyg KLQand for anx2Le_K (a generalization of the primitive element theorem from the classical field theory). In Lemma 2 and Corollary 2 we give a general Archimedean frame for the classical idea of Krasner (Krasner's Lemma) [1] or [7]. We apply this new ideas to prove that iff;gare locally constant and nonconstant mapping fromC(GK) such thatgis ``sufficiently close'' tof;thenC[f]C[g] (see Theorem 8). Then we apply this last result in order to get some information on the closedK-e subalgebras ofC(G_K) (see Corollary 3, Proposition 2 and Proposition 3).

2. Let Qbe the rational number field and let C be a fixed complex number field,QC. LetQbe the algebraic closure ofQinCand letK,

QKQbe a fixed subfield ofQ. If [K:Q]51, usuallyKis called an algebraic number field. In this paper we call any subfield ofQan algebraic number field. LetG_KˆGal(Q=K) be the absolute Galois group ofK. We denote byGˆGal(Q=Q) the absolute Galois group ofQ.Gis a compact totally disconnected group relative to the Krull topology on it [1]. The Krull topology on G_KGis exactly the topology ofGinduced toG_K. IfG_K is finite, the Krull topology becomes the discrete topology.

Let us fix a tower of subfields:

K ˆK₀ K₁K₂. . .K_n . . .Q;

such that all extensionsKn=Kare normal and finite and[¹_nˆ0KnˆQ. Then the Galois groupsHn ˆGal(Q=Kn);nˆ0;1;2;. . .are closed and open and form a fundamental system of neighborhoods of the identity automorphism e of G_K. Let denote by j:j the usual absolute value on C. For any s2GK;x ! js(x)j ˆ jxj_sis an Archimedean absolute value onQ. Letebe the conjugation automorphism ofQ. Ife2GKthenj:j_esˆ j:j_s. So any two automorphismss;m2G_Kgive rise to one and the same topology onQ(j:j_s andj:j_mare dependent) if and only ifsˆem. For anyx2Qone considers:

kxk_Kˆmaxfjs(x)j:s2G_Kg:

It is not difficult to prove the following result.

LEMMA 1. The map x ! kxk_K is a norm onQand(Q;k:k_K)is a K- normed algebra, i.e.:

i) kxk_K ˆ0 if and only ifxˆ0;for anyx2Q, ii) kxyk_K kxk_Kkyk_K;for anyx;y2Q, iii) kx‡yk_K kxk_K‡ kyk_K;for any x;y2Q, iv) kaxk_Kˆ jajkxk_K;fora2K;x2Q, and

v) k1k_Kˆ1:

We call this norm theK-spectral normonQ:It extends toQthe usual absolute value ofK. In general, in ii), one has no equality. For instance, if KˆQ and xˆ1‡ 

p2

;yˆ1 

p2

2Q, then kxyk_Kˆ16ˆ(1‡ 

p2 )² ˆ

ˆ k1‡ 

p2

k_Kk1 

p2

k_K. Taking count of the Artin-Schreir theory [6] we see that theK-spectral normx ! ks(x)k_Kis an absolute value onQequal to the usual absolute value if and only ifKis a ``real closed'' subfield inQ, i.e. if K(i)ˆQ. In this last caseG_K Z₂ and this case is the only case whenG_Kis finite. LetQe_Kbe the completion ofQrelative to theK-spectral norm. The topological closureKeofKinQe_KisRorCwheneverKR\Q or not. HenceKis dense inC(relative toj:j) if and only ifK6R\Q. In

generalQeKhas zero divisors. It is a field if and only ifKis a ``real closed'' subfield inQ. In all the other casesQeKhas zero divisors. But alwaysQeKis aK-Banach algebra.e

Let us now consider theC-Banach algebra

C(G_K)ˆ ff :G_K !Cjf continuousg;

with the usual sup-norm:kfk ˆsupfjf(s)j:s2GKg. Since GK is a com- pact group, this last quantity is always finite. For anya2Qone defines :

W_a:G_K !C;W_a(s)ˆs(a):

The mapW_ahas only a finite number of values namely, the image ofW_ais the orbit of a (relative to GK), i.e. the set of all conjugates of a in Q. If a2K;W_a(s)ˆafor anys2GK. So we can embedKe intoC(GK) by asso- ciating toa2K, the constant mapW_a inC(G_K).

PROPOSITION1. The mapW:Q ! C(G_K)defined byW(a)ˆW_a;W_a(s)ˆ

ˆs(a)is an isometry and aK-algebra morphism.So it is aK-embedding of (Q;k:k_K)into(C(GK);k:k).

PROOF. It is clear that W_a‡W_bˆW_a‡b; W_aW_bˆW_ab and lW_aˆW_la for any a;b2Q and l2K. Now kak_Kˆmaxfjs(a)j:s2G_Kg ˆ

ˆsupfjW_a(s)j:s2GKg ˆ kW_ak and the proof is complete. p For some later applications we fixa2Qand as02G_K. We are going to describe W_a¹(s₀(a)). If a2K, W_a¹(s₀(a))ˆG_K. If a2= K, let us take LˆK(a)Qand let us consider H_L the corresponding subgroup ofL:

H_Lˆ fm2G_K:m(a)ˆag. ThenW_a¹(s₀(a))ˆs₀H_L, the left coset ofs₀in GK=HL. Since HL has the index mˆdegKa;HL is closed and open in GK: Hence s0HL is also closed and open. It follows that W_a is a continuous map which has the same value on each s_iH_L, where

G_KˆeH_L[s₂H_L[. . .[s_mH_L is the partition of G_K, corresponding to

the left classes ofG_K=H_L:

COROLLARY 1. There is one and only one isometry and K-algebrae morphismWwhich extend by continuityWtoQe_K:W(f)ˆ lim

n!1W_an, where f 2Qe_K, f ˆ lim

n!1a_n relative tok:k_Kanda_n2Q;nˆ1;2;. . ..

REMARK 1. Since ks(a)k_Kˆ kak_K for any s2GK and a2Q, any s2GKis a continuous automorphism ofQrelative tok:k_K. This is the main reason to consider the K-spectral norm on Q instead of usual absolute

valuej:j. Iffang_nis a Cauchy sequence inQrelative tok:k_K, thenfs(an)g_n is also a Cauchy sequence inQrelative to the same norm. So the element s(Aˆ lim

n!1an)ˆ lim

n!1s(an) does exist in Qe_K. We denote it by s(A). If f_n!^{k k:} f inC(G_K) thenkfk ˆ lim

n!1kf_nk, soks(A)k_Kˆ kAk_Kfor anyA2Qe_K. MoreoverkAk_Kˆ lim

n!1k kan _K.

Sometimes we identifyQe_K with its imageW(Qe_K) inC(G_K).

DEFINITION1. (see also [13]). Assume thate2G_K. A mapf :G_K !C is said to be symmetric iff(es)ˆf(s), wheref(s) is the usual conjugate of f(s) inCandsis any element ofG_K.

The K-subalgebra of all symmetric maps ofe C(GK) is denoted by Csym(GK). It is closed inC(GK), so it is aK-Banach subalgebra ofe C(GK).

THEOREM1. Let K be a proper subfield ofR\Q.ThenQeK isK-iso-e morphic and isometric withCsym(GK).

PROOF. Sincee2G_Kin this case (e2G_Kif and only ifKR\Q), we can use the same technique as in [13]. However we prefer here a direct proof, which could be useful in other future researches. First of all we must remark thatW_a, for anya2Q, is a symmetric continuous map. Since any x2QeK is a limit ofxn 2Qwith respect to k:k_K, one has that W_xn!W_x in C(G_K); where W_x(s)ˆs(x) (see Remark 1). One can use the fact that the conjugation automorphism e is continuous relative to j:j on C. So

n!1lim W_xn

(es)ˆe lim

n!1W_xn(s)

ˆ lim

n!1W_xn(s), i.e.W_x2 C_sym(G_K). We prove now that any element f of C_sym(G_K) can be approximated by ``algebraic numbers''W_x, wherex2Q. Letf b e inC_sym(G_K) and lete>0 be a small positive real number. LetHbe a finite index subgroup ofGK;e2= Hand such that ifGK ˆs1H[. . .[smH; s1ˆe, is the corresponding partition ofG_Kinto left cosets, one hasjf(s) f(m)j5^e₃whenevers;m2s_iHfor an i2 f1;2;. . .;ng. Let us writel_iˆf(s_i);foriˆ1;2;. . .;n. Sincee2G_Kwe can divides₁;. . .;s_minto two classes relative to the subgroupfe;eg:

fs1;. . .;smg ˆ fs1;. . .;st;st‡1ˆes1;. . .;s2t ˆestg; mˆ2t Since f(es_j)ˆf(s_j);jˆ1;2;. . .;t, one has that l_t‡hˆl_h for any

hˆ1;2;. . .;t. We focus only on the points l₁;l₂;. . .;l_t in the com-

plex plane C. Since K6ˆR\Q, the codimension of K in R\Q is 1 (see Artin-Schreier theory [6]). So one can take a subfield L of Q;LK;[L:K]ˆlt. We can enlarge L to L⁰Q such that iˆ 

p 1

2L⁰andL⁰=Kbe a normal extension. So this lastL⁰is dense inC.

We can find x1;x2;. . .;xt2L⁰ such that js_i(x_i) l_ij5e

3;iˆ1;2;. . .;t.

One can now use the Approximation Theorem (see [1]) to find x2L⁰ such thatjx x_ij_si ˆjs_i(x) s_i(x_i)j5e

3for anyiˆ1;2;. . .;t. This means thatjsi(x) lij52e

3 for anyiˆ1;2;. . .;t. Butjesi(x) lij52e

3 implies that js_i(x) l_ij52e

3 for all iˆ1;2;. . .;m. Let us take s2G_K and choose j2 f1;2;. . .;mgsuch thatjf(s) l_jj5e

3. Thenjf(s) s_j(x)j5efor thisj.

We can enlargeL⁰such thatHˆGal(Q=L⁰) be good enough for oure>0.

Thens2s_jHmeans thats(x)ˆs_j(x) b ecausex2L⁰. Sojf(s) s(x)j5e orjf(s) W_x(s)j5efor anys2GK. Hencekf W_xk5e for anx2Qand

the theorem is completely proved. p

REMARK 2. IfKˆR\Q, then G_K ˆ fe;eg because Qˆ(R\Q)(i).

So C(G_K)ˆCC and C_sym(G_K)ˆCˆ f(a;a)ja2Cg CC. But QeKˆR[i]ˆC. Hence Theorem 1 is true even if K ˆR\Q. If KR\Q and [R\Q:K]51, then [Q:K]51 and K would be real closed in Q. Then QˆK(i)ˆ(R\Q)(i) and [Q:K]ˆ[Q:R\Q]ˆ2 which impliesKˆR\Q(see Artin-Schreier theory [6]).

THEOREM2. Let K6R\Qbe a subfield ofQ.ThenQeK is isometric and isomorphic as aC-algebra withC(G_K).

PROOF. In fact K6R\Q means that e2= GK. So the ideas of [13]

cannot be applied in this case. But in this caseKis dense inQand inC. Since any continuous function f :G_K !C can be uniformly approximated by locally constant functions with a finite number of values, one has only to prove that such a mapg:G_K !C;g(s)ˆl_i for anys2s_iH;i2 f1;2;. . .;mg, whereGK ˆs1H[s2H[. . .[smH;s1ˆe;[GK :H]ˆm, is a partition of GKrelative to a closed and open subgroupHofGK, can be approximated by elementsW_x;withx2Q. Lete>0 be a small real positive number such that

D(l_i;e)ˆ fy2C:jy l_ij5eg;iˆ1;2;. . .;m

be a disjoint set of balls centered in l_i. Since K is dense in C, we can take x₁;x₂;. . .;x_m2K and x_i2D l_i;e

2

for all iˆ1;2; ::;m. Let Lˆ fy2Q:m(y)ˆy for anym2Hg be the fixed field of H. Since

e2= G_K; fs₁;s₂;. . .;s_mg give rise to independent absolute values

:

j j_s1; :j j_s2;. . .; :j j_smonL:We use now the Approximation Theorem [1] and

find x2L such that

jx xij_siˆjsi(x) si(xi)j5e

2 for anyiˆ1;2;. . .;m:

Buts_i(x_i)ˆx_ibecausex_i2K, so we get js_i(x) x_ij5e

2;iˆ1;2;. . .;m:

This last inequality together withjx_i l_ij5e 2imply js_i(x) l_ij5e for any iˆ1;2;. . .;m:

Let us take m2G_K and j2 f1;2;. . .;mg such that m2s_jH, then m(x)ˆs_j(x) and g(m)ˆl_j. Hence jW_x(m) g(m)j5e for anym2G_K; so kW_x gk5e for an x2Qand the proof of the theorem is complete. p THEOREM 3. Let KR\Q be a subfield of Q (e2GK).Then C(GK)QeK[ei], whereei is the constant functionei(s)ˆiˆ 

p 1

for any s2G_K.

This is an isomorphism ofR- algebras.If on Qe_K[ei]we consider the componentwise topology induced by Qe_K, this last isomorphism is also a homeomorphism.

PROOF. The idea of the proof is the same like in [13]. For anyf 2G_Kwe consider:

f₁(s)ˆRe f(s)‡Re f(es) 2

Im f(s) Im f(es)

2i ; and

f₂(s)ˆRe f(s) Re f(es)

2i ‡Im f(s)‡Im f(es)

2 ;

where Re(a‡ib)ˆa and Im(a‡ib)ˆb. It is not difficult to see that f1;f22 Csym(GK) andf ˆf1‡eif2. p The following three theorems are the natural generalizations of Theo- rems 1, 2 and 3. We do not give the proofs of these theorems here because the ideas of the proofs are the same like those used to prove the above last three theorems.

THEOREM4. Assume e2GKand let LQ;such that L=K be a sub- extension ofQ=K.LetLeKbe the completion of L relative to the K-spectral norm.Let Hˆ fs2G_K:s(x)ˆx for all x2Lg be the subgroup of G_K which fixes L.Then the natural embedding W_H:L ! C(G_K=H), where

GK=H is the quotient topological space of the left classes of H in G, W_H(x)ˆW_x, whereW_x(sH)ˆs(x)can be extended to aK-isomorphism ande isometryW_H:Le_K ! Csym(G_K=H).

THEOREM5. If e2= GK, then the same isomorphism and isometry like that in Theorem 2, give rise to a K-Banach algebra isomorphism (iso-e metry)Le_K C(G_K=H).

THEOREM6. If e2G_K, then the same isomorphism and isometry like that in Theorem 2, give rise to a K-Banach algebra isomorphism (iso-e metry)C(GK=H)eLK[ei].

Theorem 2.2 of [13] can be rewrite in this new general context. We do not prove it because the ideas of the proof are the same like in [13].

THEOREM7. For any LQ;KLQ,Le_Khas a topological generic element x, i.e. Le_KˆK[x].In particular, there exist zg 2Qe_K such that QeKˆK[z].g

This last result can be considered as a generalization of the classical primitive element theorem from field theory. It will be useful for future research on the structure of the groupAutcont(QeK):

3. LetKbe a subfield ofQand letC(GK) be theC-Banach algebra of all continuous functions defined onG_K ˆGal(Q=K), with values inC. For any f 2 C(G_K) one denotes byv(f)ˆinffjf(s) f(m)j:f(s)6ˆf(m);s;m2G_Kg orv(f)ˆ0 iff is a constant function. It is easy to see thatv(f)>0 if and only iff has a finite number of distinct values and at least two (f is not constant). Here is a Krasner type lemma [1] forC(GK):

LEMMA 2. Let f;g2 C(G_K)such that f is a locally constant map and v(f)>0.Let us assume that kf gk5v(f)

2 and that g(s)ˆg(m) for s;m2G_K.Then f(s)ˆf(m).

PROOF. Let us suppose thatf(s)6ˆf(m). Thenv(f) jf(s) f(m)j jf(s) g(s)j‡jg(m) f(m)j 2kf gk5v(f), a contradiction. So f(s)ˆ

ˆf(m). p

Using this result one can obtain an Archimedean version of Krasner Lemma [1] forQ.

COROLLARY 2. Let a;b2Q, a2= K such that ka bk_K5v(a) 2 , where v(a)ˆinffja_j a_ij:a_j6ˆa_iganda_i;a_jare all the conjugates ofaˆa₁inQ over K.Then K(a)K(b).

PROOF. Let W_a and W_b be the corresponding locally constant maps in C(GK) of a and b respectively (W_a(s)ˆs(a);s2GK, etc.). Then v(W_a)ˆinffjs(a) m(a)j:s(a)6ˆm(a);s;m2GKg ˆv(a). Since kW_a W_bk ˆ

ˆ ka bk_K5v(W_a)

2 we can apply Lemma 2 and obtain that W_b(s)ˆW_b(m)

ˆ)W_a(s)ˆW_a(m). Let us takesˆe, the identity ofGK. This last implication becomesm(b)ˆb)m(a)ˆaorm2Gal(Q=K(b)))m2Gal(Q=K(a)), i.e.

K(a)K(b). p

This last result can be generalized as follows.

THEOREM 8. Let f;g be two locally constant maps in C(GK) with v(f)>0(f not constant).LetVCbe a subfield ofCsuch that f(s)and g(s)2V for all s2GK.We assume that kf gk5v(f)

2 .Then V[f] V[g] in C(G_K).Here V[f]ˆ fa₀‡a₁f‡. . .‡a_tf^tja₀;a₁;. . .;a_t 2Vg and fⁱ(s)ˆ[f(s)]ⁱ for any s2GK and i2N.Moreover, if kf gk51

2minfv(f);v(g)g and ifv(g)>0, thenV[f]ˆV[g].

PROOF. SinceGK is a compact groupf andghave a finite number of values. Let m be the number of distinct values of g:g(s1)ˆl1;

g(s₂)ˆl₂;. . .;g(s_m)ˆl_m2V. Then we can solve the linear system ofm

equations andmunknownsA₀;A₁;. . .;A_{m 1}:

f(s₁)ˆA₀‡A₁l₁‡A₂l₁²‡. . .‡A_{m 1}l₁^{m 1} f(s2)ˆA0‡A1l2‡A2l22‡. . .‡Am 1l2m 1

: : : : : :

: : : : : :

: : : : : :

f(s_m)ˆA₀‡A₁l_m‡A₂l_m²‡. . .‡A_{m 1}l_m^{m 1} The determinant D of the system is equal to Q

i5j(li lj)6ˆ0. So A_iˆD_i

D 2V for all iˆ0;1;. . .;m 1. Now, if g(s)ˆg(s_i) for an

iˆ1;2;. . .;m, thenf(s)ˆf(s_i) (see Lemma 2) for the samei. Hence, for

thesefAig;iˆ1;2;. . .;mwe can write:

f(s)ˆA₀‡A₁g(s)‡A₂g²(s)‡. . .‡A_{m 1}g^{m 1}(s); i:e:

f ˆA₀‡A₁g‡A₂g²‡. . .‡A_{m 1}g^{m 1}; or

f 2V[g]. ThereforeV[f]V[g]:The last statement is simply obtained be

changingf andg;one to each other. p

COROLLARY 3. Let B be a C-subalgebra of C(G_K) and let f be a nonconstant locally map inB, the topological closure of B ine C(G_K).Then f 2B.

PROOF. SinceBis dense inB, lete gb e inBsuch thatkf gk5v(f) 2 . By Theorem 8 we obtain thatC[f]C[g]B, i.e.f 2B. p PROPOSITION2. LetKLQbe subfields of the algebraic closure of Q.LetQe_Kbe theK-spectral closure ofQand letLe_KQe_Kthe topological closure ofLinQe_K(relative to theK-spectral norm).ThenLe_K\QˆL.

PROOF. Let x2LeK\Q. If x2K; then x2L: If x2= K; then v(x)ˆminfjx_j x_ij:x_j6ˆx_igwherex_iare all theK-conjugates ofxˆx1

in Q, is not zero. Let y2L such that kx yk_K5v(x)

2 . Then K(x) K(y)L(see Corollary 2). Hencex2Land soLeK\QˆL. p We recall that the mappingW:Q! C(GK);defined byW(x)ˆW_x;where W_x(s)ˆs(x) for any x2Qands2G;is an isometric K-embedding ofQ intoC(G_K):

PROPOSITION 3. Let A be a closed K-subalgebra of C(G_K).Then A\W(Q)is a subfield ofW(Q).

PROOF. Let BˆA\W(Q) and let x2Bnf1g. Since QB we may suppose thatkxk_K51. SinceC(GK) is a Banach algebra overC(see [5] or [18]) and because A is closed, the series 1‡x‡x²‡. . .ˆy2A. But yˆ 1

1 x, so 1 x is invertible inB(because inW(Q) it is !). So we just proved that for any x2B;x6ˆ1;1 x has an inverse inB. Letm b e a maximal ideal ofBand letz6ˆ0;z2m. Thenzˆ1 (1 z) has an inverse inB, i.e.mˆB, thusBis a subfield ofW(Q). p Propositions 2 and 3 can be viewed as the first steps to a Galois theory in Qe_K:

Acknowledgments. We express our gratitude to the referee for his patience in reading our paper and for his (or her) opinions which directly led us to improve the first version of the article (in particular Theorem 8).

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Manoscritto pervenuto in redazione l'11 novembre 2007.