On the closed subfields of Q¯~p

Sever Achimescu; Victor Alexandru; Corneliu Stelian Andronescu

doi:10.1515/math-2016-0032

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On the closed subfields of Q¯~p

Sever Achimescu , Victor Alexandru und Corneliu Stelian Andronescu

Veröffentlicht/Copyright: 25. Mai 2016

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$Open Mathematics$

Aus der Zeitschrift Open Mathematics Band 14 Heft 1

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 v_p. 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 v_p 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 Q_p be the field of p-adic numbers and let C_p be the completion of a fixed algebraic closure Q_p of Q_p with respect to the unique extension of the p-adic valuation v_p from Q_p to Q_p. In [1, 2] there were proved some results for closed subfields L of C_p, namely L=Qpx~ for certain x ∈ L called generic elements. Thus if L/Q_p is infinite, L is isomorphic (both algebraically and topologically) to a completion of a polynomial ring Q_p[x] with respect to a certain extension to Q_p[x] of the p-adic valuation on Q_p. 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 v_p 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 C_p 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 v_p the p-adic valuation as defined in [4]. Let υ be a fixed valuation on Q which extends v_p. We denote G = Gal(Q = Q) the absolute Galois group of Q and for each σ ∈ G let συ be the valuation defined on Q as follows:

σv¯x=v¯σ−1x,∀x∈Q¯.

Note that for each valuation υ′ Non Qwhich extends v_p there exists σ ∈ G such that υ = συ.

Let Q_p be the field of p-adic numbers and let Q_p> ⊇ Q_p denote a fixed algebraic closure of Q_p. The unique extension to Q_p of v_p will also be denoted v_p; its unique extension to Q_p will be denoted υ_p. Let C_p be the completion of Q_p with respect to υ_p. Finally, the unique extension of υ_p to C_p will also be denoted υ_p. 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 υ_p at Q. Moreover, if we consider Q as a subfield of Q_p, its topological closure in C_p with respect to υ is C_p. We will denote by υ_p the unique extension of υ from Q to C_p.

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

w(x)=infσ∈G(v¯(σ−1(x))=infσ∈G(σv¯)(x), ∀x ∈ Q¯.

It is easy to check the following:

w(x) = +∞ iff x = 0;
w(x + y) ≥ inf (w(x), w(y)), ∀x, y ∈ Q;
w(xy) ≥ w(x) + w(y), ∀x, y ∈ Q and w(xⁿ) = nw(x), ∀n ≥ 1.

The pseudovaluation w is said to be the spectral extension of v_p 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 x ∈ Q¯~p then x = lim_n_→∞x_n, where (x_n)_n is a Cauchy sequence in (Q, w), that is w~(x_n₊₁–x_n) = inf_τ_∈G(υτ(x_n₊₁–x_n)) → +∞.

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

Since w is an extension of v_p it follows that (Q_p, vp) is a valued subfield of the ring (Q¯~p, w~);
Q¯~p ⊆∏_σ∈G C_p^σ, where C_p^σ = C_p, ∀_σ ∈ G so that x ∈ Q¯~p ⇒ x = (x_σ)_σ∈G, x_σ ∈ Cpσ, where x_σ = lim_nσ(x_n) with respect to υ, with the operations "+" and "." componentwisely defined. We have that w(x) = inf_σ_∈G(υ(x_σ));
Q¯~p is a Banach algebra over Q_p, being complete with respect to the following norm (the spectral norm):
‖x‖=(1p)w~(x)=supσ∈G(|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, Q_p⊆K ⊆ C_p. Then there is an element z ∈ C_psuch thatL=Qpz~(the completion ofQ_p[z] with respect toυ_p).

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

Theorem 2.3

If F is a closed subfield of C_pthen F ⋂ Q_pis dense in F.

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

Theorem 2.4

IfQpalg = Kis the topological closure of Q inQwith respect toυ, a fixed extension ofv_ptoQand L is a maximal subfield ofQsuch that the restriction ofυto L is the unique extension of v_p to L, thenQ = K · L, K ⋂ L = Q and L is dense inQwith respect toυ.

Corollary 2.5

Let Q ↪ L ↪ Q ↪ Q_p ↪ C_pbe the canonical inclusions and let us denoteυthe unique continuous prolongation ofυtoQ_pand to C_p. Then the closure of L with respect toυis C_p.

Remark 2.6

A canonical embedding ofQinQ¯~pis described as follows: Let (x_n)_n≥0be a sequence inQwhich converges with respect to w. Then for any σ ∈ G the sequence (σ(x_n))_n≥0converges with respect toυ. Put x = lim_n_→∞x_n with respect to w and put x_σ = lim_n_→∞σ(x_n) with respect toυin a fieldCpσ = C_p. Therefore x = (x_σ) _σ∈G. In particular if x_n = α ∈ Q, ∀n ≥ 0 then x_σ = lim_n_→∞σ(α) = σ(α) that is the embedding ofQinQ¯~pis

α→(σ(α))σ∈G.

3 Closed subfields in Q¯~p

Proposition 3.1

Let L be a subfield ofQsuch that the restriction of the valuationυ_pto L is the unique extension of v_p to L. LetL~be the completion of L with respect to the pseudovaluation w. ThenL~is a field.

Proof. We prove that any nonzero element of the closed subring L~ of Q¯~p is invertible in L~. Let y ∈ L~, y ≠ 0, y = lim_n_→∞α_n (with respect to w), α_n ∈ L. Since ˛n 2 L and the restriction of υ to L is the unique extension of υ_p to L it follows that υ(σ(α_n)) = υ(α_n), ∀σ ∈ G. Thus all the components of y are nonzero, that is y = (y_σ)_{σ ∈ G} ith 0 ≠ y_σ ∈ Cpσ C_p. In fact y_σ ∈ L^ ⊆ C_p where L^ denotes the completion of L with respect to any of the valuations συ_p (which coincide on L). Since y_σ = lim_n→∞σ(α_n) with respect to υ_p ∀ σ ∈ G, it is clear that for n large enough we have α_n ≠ 0, thus we may assume that α_n ≠ 0, ∀n and υ(σ(α_n)) is uniformly bounded. It follows that the sequence (αn−1)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 R ⊆ Q¯~pbe a closed subfield. Then the p-adic valuation υ_p onQhas a unique exension to the subfield L =Q ∩ R

Proof. Let us assume by contradiction that υ_p does not extend uniquely to L. Then there exists σ ∈ G 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 = (x_σ)_σ∈_G ∈Q¯~p, x ≠ 0. Since the mapping ϕ: G → R, ϕ(σ) = υ(x_σ) is continuous and G is a compact space relative to the Krull topology, it follows that there exists σ₀ ∈ G such that w(x) = inf_σ_∈Gυ (x_σ) = υ(xσ₀)

We will make use of the following

Lemma 3.4

Let x = (x_σ)_σ∈G ∈Q¯~psatisfying w(x) ≥ 0. Then there exists q ∈ N {0} such that the sequence x; x^q; :::; x^qn; ::: converges inQ¯~pwith respect to w and denoting its limit by y = (y_σ)_σ∈Gwe have:

v¯p(xσ)>0⇒yσ=0

and

v¯p(xσ)>0⇒yσ≠0

Proof. Let x = lim_n_→∞x_n with respect to w, x_n ∈ Q. Thus x_σ = lim_n_→∞σ(x_n) with respect to υ. It follows that there exists a positive integer n₀ such that for all n ≥ n₀ and for all σ ∈ G, σ(x_n) and x have the same image in the residue field of υ; which is F_p. Let us denote K_n₀ the normal closure of Q(x_n₀ ). We notice that these images belong to the residue field of the restriction of υ to K_n₀ . This residue field is a finite field of the form F_q0 for a p-power qo. Now for each σ ∈ G we have either υ(x_σ) > 0 thus υ(xσq0) 0 or υ(x_σ) = 0 = w(x) thus υ(xσq0) = 0 but υ(xσq0 - x_σ > 0. Thus υ(xσq0 - x_σ > 0 for all σ ∈ G. It is known that the function ϕ_x: G ↦ R ∪{∞} defined Φ_x(σ) = υ (xσq0 – x_σ) is continuous on G being considered the Krull topology. Since ϕ_x(G) ⊆ (0, + ∞) and G is compact with respect to the Krull topology it follows that there exists M ∈ (0; 1) such that υ(xxσq0 – x_σ)≥ M for all σ ∈ G. Since υ(qo) ≥ 1, it follows that for all σ ∈ G we have that xσq0 = x_σ + z_σ with z_σ ∈ C_p, υ ≥ M and we also have

v¯(xσq02−xσq0)=v¯((xσ+zσ)qo−xσq0=v¯(q0xσq0−1zσ+...+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(xσq2−xσq) ≥ 1 + M >2 and inductively we obtain that υ(xσqs−xσq(s−1)) > s – 1 for all s ≥ 2 and for all σ ∈ G (by using the fact that for any q^s, a power of p, we have that p divides q^s!/i!(q^s – i) for all 1 ≤ i ≤ q^s_–1.) Thus the sequence x; x^q;x^q2 ; ::: converges with respect to w and the conclusion follows.

Proposition 3.5

Let R be a closed subfield ofQ¯~p. Let x = (x_σ)_σ∈G ∈ R. Then

w(x)=v¯(xσ), ∀σ∈G

Proof. Let x = (x_σ)_σ∈G ∈ R, x ≠ 0. 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(x^t₎ = r thus w(x^tp^–r) = 0. Put z = x^tp^–r, so that w(z) = 0. Note that w(x) = υ(x_σ), ∀σ ∈ G if and only if w(z) = υ(z_σ), ∀σ ∈ G; therefore it suffices to prove the proposition for z.

In other words, we start with an arbitrarly fixed x = (x_σ)_σ∈G ∈ R and we assume without loss of generality that w(x) = 0.

Let us assume by contradiction that there exists σ₀ ∈ G such that υ(x_σ₀) > 0 (we may assume positiveness by eventually replacing x by x^–¹). Lemma gives a non-zero element in R, y = (y_σ)_σ∈G with y_σ₀ = 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 ofQ¯~p, which contains all the negative powers of p, and for whichυ(x_σ) = w(x), ∀σ ∈ G and ∀x = (x_σ)_σ∈G ∈ R. Then R is a field.

Proof. Let x ∈ R, x ≠ 0. We consider three cases.

Case 1. w(x – 1) > 0

In 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) = 0

Let α_n ∈ Q be a sequence converging to x with respect to w. Thus x_σ = lim_n_→∞σ(α_n) with respect to υfor all α ∈ G. Let n_o ∈ N such that w(x – α_n) > 0 for all n ≥ n₀. Let us denote α_n the residual image of α_n. Then for all n ≥ n_o we have that α_n belongs to a fixed field F_{q_n₀} and α_n ≠ 0. Thus it follows that αn−qn0−1 = 1 that is υ_p(αn−qn0−1 –1) > 0 thus, by making n →1, we obtain υ_p(xeqn0−1−1)> 0. It follows that υ((xσqn0−1−1) –1) > 0 for all α ∈ G thus w(x^q_n₀–1 – 1) > 0. By applying the first case we obtain that therefore x is invertible in R.

Case 3. w(x) ≠ 0

Let x ∈ R –{0} and let a ∈ Z, b ∈ N such that w(x) = ab. Then w(x^bp^a) = 0 and x^bp^a ∈ R since p^–a ∈ R. By applying the second case we obtain that x^bp^a 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 ofQ¯~p. For each σ ∈ G let us define f_σ : R → C_p, f_σ(x) = x_σ. Then f_σ is a field homomorphism satisfyingυ o f_σ = w. Thus the restriction of w to R is a valuation and the fields R and f_σ(R) are isomorphic as valued fields. Moreover, f_σ induces an increasing (with respect to inclusion) function from the set of the closed subfields ofQ¯~pto the set of the closed subfields ofC_p.

Corollary 3.8

Q¯~pcontains a maximal closed subfield R isomorphic (both algebraically and topologically) toC_p.

Proof. Let Q ⊆ L ⊆ Q ⊆ 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 C_p 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 C_p. 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 K_e = f_e(R) and let z_e ∈ C_psuch that K_e = Qpze~as in Theorem1 of [5]. Let L be as in the proof of the above corollary and let (α_n)_nbe a sequence in L such that z_e = lim_nα_n with respect toυ. For each σ ∈ Gal(Q/Q) we denote z_σ = lim_nσ(α_n) and K_σ = f_σ(K) and we have that Qpzσ~. Let z = (z_σ)_σ∈G. Then the ringQ[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

LetQpalg = K be the topological closure ofQinQwith respect toυ N, a fixed extension of υ_p toQand let R be the maximal subfield ofQ¯~pas in the previous Corollary. Then the subring_K~and R generated byK~and R is dense in (Q¯~pw~).

Proof. Let R = L~, where L denotes a maximal subfield of Q to which υ_p 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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Received: 2016-1-13

Accepted: 2016-5-10

Published Online: 2016-5-25

Published in Print: 2016-1-1

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