METRIC PROPERTIES OF THE DEGREE FUNCTION OVER A p-ADIC FIELD

METRIC PROPERTIES OF THE DEGREE FUNCTION OVER A p-ADIC FIELD

MARIAN V ˆAJ ˆAITU and ALEXANDRU ZAHARESCU

Letpbe a prime number,Qpthe field ofp-adic numbers,Ka finite field extension ofQp, andK a fixed algebraic closure of K. We discuss some metric properties of the degree function deg_K:K→N,α→deg_K(α) = [K(α) :K].

AMS 2000 Subject Classification: 11S99.

Key words: p-adic number, degree function, metric invariant.

1. INTRODUCTION

Letpbe a prime number,Qp the field ofp-adic numbers,K a finite field extension ofQ_p,K a fixed algebraic closure ofK, andC_p the completion ofK with respect to the thep-adic absolute value| · |onQ_p, normalized by|p|= ¹_p. In this paper we raise some problems concerned with metric properties of the degree function deg_K : K → N, given by α → deg_K(α) = [K(α) : K], the degree ofαoverK. A neat result (see Theorem 1 from [7]) states that for any ballB inK, the set of degrees of elements fromB, i.e. {deg_Kα :α∈B}, is an arithmetic progression of the form{dm:m≥1}. Heredis the smallest degree overK of an element fromB,d= min{deg_K(α) :α∈B}. In the light of this result, a natural question that arises is whether such a clean characterization of the set of degrees {deg_K(α) : α ∈ B} can still be obtained in case B is replaced by other subsets ofK. In this paper we will restrict ourselves to the case whereB is replaced by a more general open subset ofK. It would be also interesting to investigate similar questions for other types of subsets ofK.

Further questions may be asked about the behaviour of the degree func- tion and its dependence on B, if B varies within a family of subsets of K. Similarly, one could vary the p-adic field K in a family of finite field exten- sions of Q_p and one may study the behavior of the degree function and its dependence on K. Let us now return to the case where K is a fixed finite field extension of Q_p, and B runs over a family of balls, or more generally over a family of open subsets ofK. We would like to connect the image ofB through the map deg_K, that is, the set of degrees of elements fromB, to other

REV. ROUMAINE MATH. PURES APPL.,52(2007),1, 111–119

invariants that naturally appear in the study of local fields. In this respect, we discuss below some connections between such questions and the theory of saturated distinguished chains of invariants over a local field.

2. NOTATION, TERMINOLOGY AND RESULTS A pair (α, β) of elements ofK is called adistinguished pair if

deg_K(α) >deg_K(β), (1)

ifγ ∈K and deg_K(γ)<deg_K(α), then|α−γ| ≥ |α−β|, (2)

and

ifγ ∈Kand deg_K(γ)<deg_K(β), then|α−γ|>|α−β|.

(3)

In other words, (α, β) is distinguished if β is an element of K of minimal degree for which |α −β| = inf{|α−γ| : γ ∈ K,deg_K(γ) < deg_K(α)}. If α ∈K then there exist (infinitely many) elements β ∈K for which (α, β) is distinguished. For, we observe that the set{|α−γ|: deg_K(γ)<deg_K(α)} has a latest element, since it is bounded e.g. byω(α) := inf|α−α|whereα runs over all conjugates ofα over K, andα = α. Moreover, deg_K(β) is a divisor of deg_K(α).

If we start with a given element α ∈ K, we may construct chains of elements fromK such that any two consecutive elements in the chain form a distinguished pair. An (l+ 1)-tuple (α0, α1, . . . , αl) of elements ofK such that α₀ =α, (α_j−1, α_j) is a distinguished pair for any j∈ {1,2, . . . , l}and α_l∈K, is said to be a saturated distinguished chain forα over K. In such a way, we obtain a finite sequence ofmetric invariants forα given by δi =|αi−α|, for 0≤i≤l.

Various properties of saturated distinguished chains have been estab- lished in [1–8].

Let now (M,≤) be a partially ordered set. A map u:M → Mis called asuperior regularization onM (see [9]) if

u(x)≤u(y) for anyx, y∈ Mwithx≤y, (4)

u(u(x)) =u(x) for anyx∈ M, (5)

and

(6) u(x)≥xfor anyx∈ M.

If instead of (6) the mapu satisfies

(7) u(x)≤xfor anyx∈ M,

then we callu aninferior regularization on M.

3. METRIC PROPERTIES OF THE DEGREE FUNCTION We use the notation from the previous sections. Let B be an arbitrary subset of K. We are interested in the image of B through the map deg_K : K→N, given byα→deg_K(α) := [K(α) :K]. For simplicity, in what follows we denote this image by deg_K(B), thus

deg_K(B) ={deg_K(α) :α∈B}.

In order to better visualize various geometrical shapes the setB may take, it is convenient to consider the topological completion Cp of K and extend the map deg_K to Cp, by letting deg_K(T) = ∞ for any element T of Cp which is transcendental over K. Any open or closed ball in K is the intersection of an open or closed ball in Cp, with K. Similarly, any open subset of K is of the form Ω∩K, for some open subset Ω of C_p. Moreover, the set under investigation, namely deg_K(Ω∩K), is left unchanged via the map Ω∩K→Ω, except for the addition of the element∞. Thus

deg_K(Ω) = deg_K(Ω∩K)∪ {∞}.

Let us denote by B the family of subsets of N = {1,2, . . .} which have the property that they contain any positive multiple of any of their elements. In other words, a subset M of N belongs to B if and only if for any d ∈ M and any m ∈ N one has dm ∈ M. Thus any element M of B is a set of natural numbers, which is a union of arithmetic progressions of the form {dm:m∈N}. We also set B_∞ ={M ∪ {∞} :M ∈ B}.Next, let us denote by U the family of open subsets of C_p. With respect to inclusion, U is a partially ordered set. For any Ω∈ U we are interested in the set of degrees, deg_K(Ω). We now construct a map, call it (∼), from U to U, as follows. Fix an Ω ∈ U. As an open subset of C_p, Ω is a union of open balls in C_p. Let us fix such an open ball B. Take an arbitrary element α ∈ B ∩K. Thus B = B(α, δ) = {z ∈ C_p : |z−α| < δ}, for some δ > 0. Choose a saturated distinguished chain forα over K, (α₀, α₁, . . . , α_l) say, where lis the length of α over K, α₀ = α, α_l ∈ K, deg_K(α_i) > deg_K(α_i+1) for 0 ≤ i ≤ l−1, and consider the finite sequence of metric invariants forαgiven byδ_i =|α_i−α|, for 0≤i≤l. We also defineδ_l+1 =∞. Thus 0 =δ₀ < δ₁ <· · ·< δ_l < δ_l+1 =∞.

Let nowr∈ {1,2, . . . , l+ 1} denote the smallest index for which δ_r ≥δ. Thusδr−1 < δ ≤δr. We set B := B(α, δr) = {z∈ Cp :|z−α|< δr}.Then, we define

Ω :=

Bopen ballB⊆Ω

B.

Clearly, eachB is an open ball in Cp or Cp itself, hence Ω is an open subset ofCp. We have the following result.

Theorem 1. Let K be a finite field extension of Qp.

(a) We have Ω ⊆ Ω and Ω = Ω for any Ω ∈ U. If Ω₁,Ω₂ ∈ U and Ω₁ ⊆Ω₂, then Ω₁ ⊆Ω₂.

(b)For any family (Ω_i)_i∈I of open subsets of Cp, ∪i∈IΩ_i =∪i∈IΩ_i. (c) The map (∼) commutes with any continuous automorphism of Cp

over Q_p, that is, for any σ∈Gal_cont(C_p/Q_p) the diagram U −−−−→ U^∼

σ



^σ U −−−−→

∼ U is commutative.

(d)For any Ω∈ U, deg_K(Ω) = deg _K(Ω).

(e) The map deg_K :U → B∞ is surjective.

Remark. Property (a) above says that the map (∼) :U → Uis a superior regularization, and (b) states that (∼) commutes with sup. Property (d) states that the map (∼) invariates the image of the degree map deg_K, and (e) characterizes the sets of natural numbers which are sets of degrees of elements from an open subset ofCp.

4. PROOF OF THEOREM 1

We first show that the map (∼) : U → U is well defined. In order to do this we need to check that for any open set Ω, any open ball B contained in Ω, and any elements α, α in B which are algebraic over K, the ball B constructed with respect to α or with respect to α is the same. In other words, let (α0, α1, . . . , αl) be a saturated distinguished chain for α over K, (α₀, α₁, . . . , α_l) a saturated distinguished chain for α over K, and consider the metric invariants

δj =|α−αj|, 0≤j ≤l; δl+1 =∞;

δ_j =|α−α_j|, 0≤j ≤l; δ_l+1=∞.

Let δ denote the radius of the ball B, and let r, s be the smallest positive integers for which δr ≥ δ and respectively δ_s ≥ δ. Then the radius of B defined with respect to α is δ_r while the radius of B defined with respect to α is δ_s. Thus in order to show that the two constructions of B, the one

with respect toα and the other with respect to α, produce the same setB, it is necessary to have δr = δ_s. Note that this condition is also sufficient, since two open balls having the same radiusδr =δ_s, centered at pointsα and respectively α, at distance |α−α| strictly smaller than the radius δr = δ_s, must coincide. So we only need to show thatδ_r=δ_s. Let us assume that this does not hold and, to make a choice, assume thatδ_r< δ_s. Recall that

|α−αr|=|αr−1−αr|=δr

and

|α−α_s|=|α_s−1−α_s|=δ_s.

Also, |α−α|< δ, |α−αr−1|=δr−1 < δ, |α−α_s−1|< δ, therefore |α_s−1− αr−1|< δ. Combining this with the inequality |α_r−1−αr|=δr≥δ, it follows that |α_s−1−αr| = δr < δs = |α_s−1 −α_s|. But (α_s−1, α_s) is a distinguished pair over K, hence δ_s = |α_s−1 −α_s| ≤ |α_s−1 −β|, for any element β ∈ K with deg_Kβ < deg_Kα_s−1. Applying this relation with β = α_r we deduce that deg_Kα_r ≥ deg_Kα_s−1. On the other hand, (α_r−1, α_r) is a distinguished pair over K, hence δr = |αr−1 −αr| ≤ |αr−1−β|, for any element β ∈ K with deg_Kβ <deg_Kαr−1. We may apply this withβ =α_s−1, which satisfies deg_Kα_s−1≤deg_Kαr<deg_Kαr−1, and get that

|α_r−1−α_s−1| ≥δ_r.

But this contradicts the inequality |αr−1 −α_s−1| < δ obtained above. To conclude one hasδr=δ_s, which completes the proof that the map (∼) :U → U is well defined.

The properties of this map stated in the theorem follow now easily. For instance, to prove part (c) of the theorem, namely that the map (∼) commutes with any automorphismσ ∈G_K = Gal_cont(C_p/K), we may take any open set Ω, any open ballB contained in Ω, any elementαalgebraic overK that lies in B, and constructB with respect toα. Then conveniently choose the element σ(α) that lies in the open ballσ(B), which is contained in the open setσ(Ω), and construct σ(B) with respect to σ(α). The elements α and σ(α) being conjugates overK, they have the same sets of metric invariants. Therefore, the ballsB andσ(B), which are constructed using these sets of metric invariants, will have the same radius. But B and σ(B) have the same radius. So,σ(B) and σ(B) are open balls of the same radius, and both contain the element σ(α). It follows that σ(B) coincides with σ(B). Further, taking the union over all the open balls B contained in Ω one finds that σ(Ω) coincides with σ(Ω), which proves part (c).

Next, to show that the map (∼) : U → U is a superior regularization, note first that for each open ballB one hasB ⊆B. By taking unions over all the open ballsB contained in an open set Ω we see that Ω⊆Ω, for any Ω ∈ U. Further, if Ω₁,Ω₂ ∈ U and Ω₁ ⊆ Ω₂, then any open ball B contained in Ω₁ will also be contained in Ω₂, so that each B that contributes to Ω₁ will also contribute toΩ₂, henceΩ₁ ⊆Ω₂. Let us show now thatΩ = Ω for any Ω ∈ U. Fix such an Ω. We know already that Ω ⊆ Ω. In order to prove the other inclusion, recall thatΩ :=

Dopen ballD⊆Ω

D. It is then enough to show that for any

open ballDcontained inΩ the (possibly larger) open ball D is also contained in Ω. We know that Ω :=

Bopen ballB⊆Ω

B, thus D ⊆

Bopen ballB⊆Ω

B. Choose an element α in D, algebraic over K, for which deg_Kα is as small as possible, and let (α0, α1, . . . , αl) be a saturated distinguished sequence for α over K. Since deg_Kα₁<deg_Kα, it follows thatα₁ does not belong toDby the choice ofα. Therefore, ifδdenotes the radius ofDandδ₁ =|α−α₁|, thenδ≤δ₁, and sinceδ₀ =|α−α₀|= 0< δ, we deduce by the construction ofD with respect to αthat the radius ofD coincides withδ1. Next, recall that α∈

Bopen ballB⊆Ω

B and choose an open ball B₀ contained in Ω for which α ∈ B₀. Let β be an element ofB₀, algebraic over K, for which deg_Kβ is as small as possible, and let (β₀, β₁, . . . , β_t) be a saturated distinguished sequence for β over K. Since deg_Kβ₁ <deg_Kβ, it follows thatβ ∈B₀by the choice ofβ. Thus, ifηdenotes the radius of B₀ and η₁ = |β −β₁|, then η ≤ η₁, and the radius of B₀ will coincide with η₁. The fact that α ∈B₀ may then be restated as saying that

|β−α|< η₁. Combining this with the fact that|β−γ| ≥ |β−β₁|=η₁for any elementγ algebraic over K which satisfies deg_Kγ <deg_Kβ, we deduce that deg_Kα≥deg_Kβ. Consequently, deg_Kα >deg_Kβ₁, and by the choice ofα it is clear that β₁ does not belong to D. So |α−β₁| ≥ δ. On the other hand, combining the inequality |β−α| < η₁ with the equality |β −β₁|=η₁ yields

|α−β₁| =η₁. Therefore, η₁ ≥ δ. We deduce that the open ball of radius δ centered atα, that isD, is contained in the open ball of radiusη1 centered at α. Since |α−β|< η1, this last ball coincides with the open ball of radiusη1

centered atβ, which is B0. So, D⊆B0. As a consequence, D ⊆B0. But B0

may be obtained fromB0 by the usual construction, taken with respect to β. SinceB0is the open ball of radiusη1 centered atβ, andη1 is one of the metric invariants ofβ over K, by the definition of B₀ this open ball is also centered

atβ and has radiusη1. In other words,B0 coincides withB0. Therefore,D is contained inB0, which in turn is contained inΩ, so D ⊆Ω. Since this holds for an arbitrary open ball D contained in Ω, we conclude that Ω = Ω. This completes the proof that the map (∼) : U → U is a superior regularization i.e., part (a) of the theorem.

In order to prove part (b), let (Ω_i)_i∈I be a family of open sets and set Ω =

i∈IΩ_i ∈ U. We need to show that Ω =

i∈IΩ_i. Since Ω_i ⊆Ω for any i ∈ I, one has Ω_i ⊆ Ω for any i, so

i∈IΩ_i ⊆ Ω. For the other inclusion, using the fact thatΩ is covered by open balls of the form B, withB open ball contained in Ω, the problem reduces to show thatB ⊆

i∈IΩ_i for each open ballB ⊆Ω. Fix such a ballBand choose an elementαinB, algebraic overK, for which deg_Kα is as small as possible. Then, reasoning as above, one sees that ifδ denotes the radius ofB and δ1 =|α−α1|, where (α0, α1, . . . , α_l) is a saturated distinguished chain forαover K, thenδ₁ ≤δ, andB coincides with the open ball of radius δ₁ centered at α. Next, since α ∈ B ⊆ Ω =

i∈IΩ_i, there isi₀ ∈I, and an open ballU contained in Ω_i0, such that α∈U. Let η denote the radius ofU and choose a real numberη₁ with 0< η₁ <min{η, δ₁}. If we denote by U1 the open ball of radius η1 centered at α, then by the definition of U1, constructed with respect to α, one sees that U1 coincides with the open ball of radius δ1 centered at α, that is, one has U1 =B. On the other hand, the inclusionsU1 ⊆U ⊆Ω_i0 implyU1 ⊆U ⊆Ω_i0. Therefore, B⊆Ω_i0 ⊆

i∈IΩ_i, and this completes the proof of part (b).

We now show that deg_K(Ω) = deg_K(Ω) for any Ω ∈ U. Fix Ω ∈ U. Since Ω⊆Ω, it is clear that deg _K(Ω)⊆deg_K(Ω). In order to prove the other inclusion, take an arbitrary positive integernwhich belongs to deg_K(Ω), and choose α ∈ Ω for which deg _Kα = n. Using the fact that the open balls B, with B open ball contained in Ω, cover Ω, choose an open ball B ⊆ Ω for whichα∈B. Letβ be an element ofB, algebraic over K, for which deg_Kβ is as small as possible. Sayd:= min{deg_Kγ :γ ∈B∩K}, so deg_Kβ =d. Let (β₀, β₁, . . . , β_l) be a saturated distinguished chain forβ and letδ₁:=|β−β₁|. ThenB coincides with the open ball of radiusδ₁ centered at β. In particular, sinceα∈B, we have|β−α|< δ1. Note that for any elementγ ofB, algebraic over K, one has deg_Kγ ≥ d. Indeed, if there exists an element γ ∈ B with deg_kγ < d= deg_Kβ, then using the fact that (β, β₁) is a distinguished pair it would follow that|β−γ| ≥ |β−β₁|=δ₁, which contradicts the assumption that γ ∈ B. We deduce that min{deg_Kγ :γ ∈ B∩K} =d. It then follows

from Theorem 1 of [7] that

deg_K(B) ={dm:m≥1}.

In particular, since α∈B we deduce that n is a multiple ofd, say, n=dm0

for somem0 ≥1. On the other hand, by Theorem 1 of [7] applied to the ball B, there exists a γ0 ∈ B for which deg_Kγ0 = dm0 = n. Since γ0 ∈ Ω, we conclude thatn∈deg_K(Ω). Thus, deg_K(Ω) = deg _K(Ω), which completes the proof of (d).

Finally, we prove part (e) of the theorem. Since any Ω∈ U is a union of open balls, and for each open ballB in this union we have deg_K(B∩K) =dN^∗ for somedby Theorem 1 of [7], and by the definition ofB_∞we have deg_K(Ω)∈ B_∞. Conversely, if we take an arbitrary element ofB_∞ and intersect it with N^∗ we obtain a subset of N^∗ which can be written as a union of a (finite or infinite) sequence of arithmetic progressions of the formdN^∗. Now, each such arithmetic progression can be realized as the set of degrees over K of the algebraic elements from a suitable open ball B (this last fact follows easily:

choose an algebraic element α,α ∈ K of degree d, for example, a root of an Eisenstein polynomial of degreed, and letεbe a positive real number satisfying 0< ε < δ₁, whereδ₁ is the first metric invariant for α; then deg_K(B) =dN^∗, whereB is the open ball of radiusεcentered atα). Taking the union of these ballsB we obtain an open set whose set of degrees overK is the given element ofB_∞. This proves the surjectivity of the map deg_K :U → B_∞and completes the proof of the theorem.

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Received 1 November 2005 Romanian Academy

“Simion Stoilow” Institute of Mathematics P.O. Box 1-764

014700 Bucharest, Romania Marian.Vajaitu@imar.ro

and

University of Illinois at Urbana-Champaign Department of Mathematics Altgeld Hall, 1409 W. Green Street

61801 Urbana IL, USA zaharesc@math.uiuc.edu