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Open Mathematics

formerly Central European Journal of Mathematics

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Volume 14, Issue 1 (Jan 2016)

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On the closed subfields of Q¯~p

Sever Achimescu
  • Corresponding author
  • Departament of Mathematics and Computer Science Bulevardul Lacul Tei nr. 122-124 Cod postal 020396 sector 2 Bucharest, Romania
  • Email:
/ Victor Alexandru
  • Department of Mathematics, University of Bucharest, str Academiei nr 14 010014 Bucharest, Romania
  • Email:
/ Corneliu Stelian Andronescu
  • University of Pitesti, Faculty of Mathematics-Computer Science, 110040 Pitesti str.Targu din Vale, nr 1, Arges, Romania
  • Email:
Published Online: 2016-05-25 | DOI: https://doi.org/10.1515/math-2016-0032

Abstract

Let p be a prime number, and let Q¯~p be the completion of Q with respect to the pseudovaluation w which extends the p-adic valuation vp. In this paper our goal is to give a characterization of closed subfields of Q¯~p, the completion of Q with respect w, i.e. the spectral extension of the p-adic valuation vp on Q.

Keywords: Complete subrings; p-adic fields; Generic elements; Spectral norms

MSC 2010: 11E95; 13A18

1 Introduction

Let p be a prime number, let Qp be the field of p-adic numbers and let Cp be the completion of a fixed algebraic closure Qp of Qp with respect to the unique extension of the p-adic valuation vp from Qp to Qp. In [1, 2] there were proved some results for closed subfields L of Cp, namely L=Qpx~ for certain xL called generic elements. Thus if L/Qp is infinite, L is isomorphic (both algebraically and topologically) to a completion of a polynomial ring Qp[x] with respect to a certain extension to Qp[x] of the p-adic valuation on Qp. In this paper we give characterization of closed subfields of Q¯~p, the completion of Q with respect to the spectral extension of the p-adic valuation vp on Q. Archimedean spectral norms and nonarchimedean spectral norms on valued fields and their completions are studied in many articles including those mentioned as references. An important application, related to the structure of the compact subsets of Cp is given in [3].

This paper contains two sections. In the first section, we quote definitions and preliminary results and we fix some notation. In the second section we give a characterization of the closed subfields of Q¯~p and some consequences.

2 Background material

Let Q be a fixed algebraic closure of the field of rational numbers Q. For a prime positive integer p we denote vp the p-adic valuation as defined in [4]. Let υ be a fixed valuation on Q which extends vp. We denote G = Gal(Q = Q) the absolute Galois group of Q and for each σG let σ υ be the valuation defined on Q as follows:

σv¯x=v¯σ1x,xQ¯.

Note that for each valuation υ′ Non Qwhich extends vp there exists σG such that υ = σ υ.

Let Qp be the field of p-adic numbers and let Qp> ⊇ Qp denote a fixed algebraic closure of Qp. The unique extension to Qp of vp will also be denoted vp; its unique extension to Qp will be denoted υp. Let Cp be the completion of Qp with respect to υp. Finally, the unique extension of υp to Cp will also be denoted υp. The corresponding absolute value is denoted by |.|p, thus |x|p=(1p)v¯p(x). Certainly we can suppose that υ is the restriction of υp at Q. Moreover, if we consider Q as a subfield of Qp, its topological closure in Cp with respect to υ is Cp. We will denote by υp the unique extension of υ from Q to Cp.

In [5, 6] there has been considered a pseudovaluation w : Q → R ⋃ {+∞} induced by vp on Q as follows:

w(x)=infσG(v¯(σ1(x))=infσG(σv¯)(x),  x    Q¯.

It is easy to check the following:

  1. w(x) = +∞ iff x = 0;

  2. w(x + y) ≥ inf (w(x), w(y)), ∀x, yQ;

  3. w(xy) ≥ w(x) + w(y), ∀x, yQ and w(xn) = nw(x), ∀n ≥ 1.

The pseudovaluation w is said to be the spectral extension of vp to Q. We denote Q¯~p the completion of Q with respect to the pseudovaluation w and let w~ be the unique extension of w to Q¯~p. If xQ¯~p then x = limn→∞xn, where (xn)n is a Cauchy sequence in (Q, w), that is w~(xn+1xn) = infτG(υτ(xn+1xn)) → +∞.

We also recall from [6, 7] the following:

  1. Since w is an extension of vp it follows that (Qp, vp) is a valued subfield of the ring (Q¯~p, w~);

  2. Q¯~p ⊆∏σG Cpσ, where Cpσ = Cp, ∀σG so that xQ¯~px = (xσ)σG, xσCpσ, where xσ = limnσ(xn) with respect to υ, with the operations "+" and "." componentwisely defined. We have that w(x) = infσG(υ(xσ));

  3. Q¯~p is a Banach algebra over Qp, being complete with respect to the following norm (the spectral norm): x=(1p)w~(x)=supσG(|xσ|);    |xσ|:=(1p)v¯p(xσ).

Remark 2.1: The spectral norm defined above is also considered in [7, 8] in a more general context.

We will make use of the following theorem proved in [1, 2]:

Theorem 2.2: Let K be a complete field, Qp K ⊆ Cp. Then there is an element z ∈ Cp such that L=Qpz~ (the completion of Qp[z] with respect to υp).

Let us recall the following result proved in [1]:

Theorem 2.3: If F is a closed subfield of Cp then F ⋂ Qp is dense in F.

We will make use of a consequence of the main theorem of [5]:

Theorem 2.4: If Qpalg = K is the topological closure of Q in Q with respect to υ, a fixed extension of vp to Qand L is a maximal subfield of Q such that the restriction of υ to L is the unique extension of vp to L, then Q = K · L, KL = Q and L is dense in Q with respect to υ.

Corollary 2.5: Let Q ↪ LQQp ↪ Cp be the canonical inclusions and let us denote υ the unique continuous prolongation of υ to Qp and to Cp. Then the closure of L with respect to υ is Cp.

Remark 2.6: A canonical embedding of Q in Q¯~p is described as follows: Let (xn)n≥0 be a sequence in Q which converges with respect to w. Then for any σG the sequence (σ(xn))n≥0 converges with respect to υ. Put x = limn→∞xn with respect to w and put xσ = limn→∞σ(xn) with respect to υ in a field Cpσ = Cp. Therefore x = (xσ) σG. In particular if xn = αQ, ∀n ≥ 0 then xσ = limn→∞σ(α) = σ(α) that is the embedding of Q in Q¯~p is α(σ(α))σG.

3 Closed subfields in Q¯~p

Proposition 3.1: Let L be a subfield of Q such that the restriction of the valuation υp to L is the unique extension of vp to L. Let L~ be the completion of L with respect to the pseudovaluation w. Then L~ is a field.Proof. We prove that any nonzero element of the closed subring L~ of Q¯~p is invertible in L~. Let yL~, y ≠ 0, y = limn→∞αn (with respect to w), αnL. Since ˛n 2 L and the restriction of υ to L is the unique extension of υp to L it follows that υ(σ(αn)) = υ(αn), ∀σG. Thus all the components of y are nonzero, that is y = (yσ)σG ith 0 ≠ yσCpσ Cp. In fact yσL^Cp where L^ denotes the completion of L with respect to any of the valuations συp (which coincide on L). Since yσ = limnσ(αn) with respect to υpσ ∈ G, it is clear that for n large enough we have αn ≠ 0, thus we may assume that αn ≠ 0, ∀n and υ(σ(αn)) is uniformly bounded. It follows that the sequence (αn1)n is Cauchy with respect to w in the complete space L~. Thus it has a limit which is also the inverse of y in L~. □

A converse of the previous proposition is also true.

Proposition 3.2: Let RQ¯~p be a closed subfield. Then the p-adic valuation υp on Q has a unique exension to the subfield L = QRProof. Let us assume by contradiction that υp does not extend uniquely to L. Then there exists σG such that υ and σ(υ) give distinct restrictions to L and therefore independent. By using a similar argument to that in [6], namely Proposition 1, page 141 from [8], it follows that the completion of L with respect to w (that is a subring of R) is not an integral domain. □

Remark 3.3: Let x = (xσ)σ∈ GQ¯~p, x ≠ 0. Since the mapping ϕ: GR, ϕ(σ) = υ(xσ) is continuous and G is a compact space relative to the Krull topology, it follows that there exists σ0G such that w(x) = infσG υ (xσ) = υ(0)

We will make use of the following

Lemma 3.4: Let x = (xσ)σGQ¯~p satisfying w(x) ≥ 0. Then there exists qN {0} such that the sequence x; xq; :::; xqn; ::: converges in Q¯~p with respect to w and denoting its limit by y = (yσ)σ∈G we have: v¯p(xσ)>0yσ=0and v¯p(xσ)>0yσ0Proof. Let x = limn→∞xn with respect to w, xnQ. Thus xσ = limn→∞σ(xn) with respect to υ. It follows that there exists a positive integer n0 such that for all nn0 and for all σG, σ(xn) and x have the same image in the residue field of υ; which is Fp. Let us denote Kn0 the normal closure of Q(xn0 ). We notice that these images belong to the residue field of the restriction of υ to Kn0 . This residue field is a finite field of the form Fq0 for a p-power qo. Now for each σG we have either υ(xσ) > 0 thus υ(xσq0) 0 or υ(xσ) = 0 = w(x) thus υ(xσq0) = 0 but υ(xσq0 - xσ > 0. Thus υ(xσq0 - xσ > 0 for all σG. It is known that the function ϕx: GR ∪{∞} defined Φx(σ) = υ (xσq0xσ) is continuous on G being considered the Krull topology. Since ϕx(G) ⊆ (0, + ∞) and G is compact with respect to the Krull topology it follows that there exists M ∈ (0; 1) such that υ(xxσq0xσ)≥ M for all σG. Since υ(qo) ≥ 1, it follows that for all σG we have that xσq0 = xσ + zσ with zσCp, υM and we also have v¯(xσq02xσq0)=v¯((xσ+zσ)qoxσq0=v¯(q0xσq01zσ+...+zσq0)min(1+M,  q0M)and 1 + M ∈(1, 2). By replacing qo by one of its powers q large enough such that qM >2 we obtain as above that υ(xσq2xσq) ≥ 1 + M >2 and inductively we obtain that υ(xσqsxσq(s1)) > s – 1 for all s2 and for all σG (by using the fact that for any qs, a power of p, we have that p divides qs!/i!(qsi) for all 1 ≤ iqs 1.) Thus the sequence x; xq;xq2 ; ::: converges with respect to w and the conclusion follows.

Proposition 3.5: Let R be a closed subfield of Q¯~p. Let x = (xσ)σGR. Then w(x)=v¯(xσ),    σGProof. Let x = (xσ)σGR, x0. Eventually replacing x by x–1 we may assume without loss of generality that w(x) ≥ 0. Now if w(x) = rt > 0 with r; t positive integers then w(xt) = r thus w(xtp–r) = 0. Put z = xtp–r, so that w(z) = 0. Note that w(x) = υ(xσ), ∀σG if and only if w(z) = υ(zσ), ∀σG; therefore it suffices to prove the proposition for z.

In other words, we start with an arbitrarly fixed x = (xσ)σGR and we assume without loss of generality that w(x) = 0.

Let us assume by contradiction that there exists σ0G such that υ(xσ0) > 0 (we may assume positiveness by eventually replacing x by x1). Lemma gives a non-zero element in R, y = (yσ)σG with yσ0 = 0. It follows that the nonzero element y of the field R cannot be invertible (see [9]), contradiction.

The converse is given by the following

Theorem 3.6: Let R be a closed ( with respect to w) subring of Q¯~p, which contains all the negative powers of p, and for which υ(xσ) = w(x), ∀σG and ∀x = (xσ)σGR. Then R is a field.Proof. Let xR, x ≠ 0. We consider three cases.Case 1. w(x – 1) > 0In this case we write x = 1 – y with w(y) > 0 and since (1 – y) (1 + y + y2 + ::: + yn + :::) = 1 it follows that x is invertible in R. We used: R is closed and 1 + y + y2 + ::: + yn + ::: converges with respect to w and its limit belongs to R.Case 2. w(x) = 0Let αnQ be a sequence converging to x with respect to w. Thus xσ = limn→∞σ(αn) with respect to υfor all αG. Let noN such that w(xαn) > 0 for all nn0. Let us denote αn the residual image of αn. Then for all nno we have that αn belongs to a fixed field Fqn0 and αn ≠ 0. Thus it follows that αnqn01 = 1 that is υp(αnqn01 –1) > 0 thus, by making n →1, we obtain υp(xeqn011) > 0. It follows that υ((xσqn011) –1) > 0 for all αG thus w(xqn0–1 – 1) > 0. By applying the first case we obtain that therefore x is invertible in R.Case 3. w(x) ≠ 0Let xR –{0} and let aZ, bN such that w(x) = ab. Then w(xbpa) = 0 and xbpaR since p–aR. By applying the second case we obtain that xbpa is invertible in R thus x is invertible in R.The statements of the following theorem are straightforward. □

Theorem 3.7: Let R be a closed subfield of Q¯~p. For each σG let us define fσ : RCp, fσ(x) = xσ. Then fσ is a field homomorphism satisfying υ o fσ = w. Thus the restriction of w to R is a valuation and the fields R and fσ(R) are isomorphic as valued fields. Moreover, fσ induces an increasing (with respect to inclusion) function from the set of the closed subfields of Q¯~p to the set of the closed subfields of Cp.

Corollary 3.8: Q¯~p contains a maximal closed subfield R isomorphic (both algebraically and topologically) to Cp.Proof. Let QLQ ⊆ be field extensions with L maximal with the property that the restriction of υ to L is the unique prolongation of υ to L. As in [5] it follows that the field L is dense in Cp with respect to υ. Thus the completion L~ of L with respect to w is a subfield of Q¯~p and L~ is isomorphic to Cp. Therefore, by Theorem5, it is a maximal subfield of Q¯~p. □

Let R be a closed subfield of Q¯~p. By applying Theorem 1 of [2] it follows the following theorem.

Theorem 3.9: Let us denote Ke = fe(R) and let zeCp such that Ke = Qpze~ as in Theorem1 of [5]. Let L be as in the proof of the above corollary and let (αn)n be a sequence in L such that ze = limnαn with respect to υ. For each σ ∈ Gal(Q/Q) we denote zσ = limnσ(αn) and Kσ = fσ(K) and we have that Qpzσ~. Let z = (zσ)σG. Then the ring Q[z] is dense in R with respect to the pseudovaluation w.

Now we use the main result from [5] in order to conclude the following theorem.

Theorem 3.10: Let Qpalg = K be the topological closure of Q in Q with respect to υ N, a fixed extension of υp to Q and let R be the maximal subfield of Q¯~p as in the previous Corollary. Then the subring K~ and R generated by K~ and R is dense in (Q¯~pw~).Proof. Let R = L~, where L denotes a maximal subfield of Q to which υp extends uniquely. By applying Theorem 3 from [5] we obtain that Q = K · L and K ∩ L = Q. Thus K~L~ contains Q therefore it is dense in (Q¯~pw~). □

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About the article

Received: 2016-01-13

Accepted: 2016-05-10

Published Online: 2016-05-25

Published in Print: 2016-01-01



Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0032. Export Citation

© 2016 Achimescu et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

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