*Let L be a subfield of* **Q** *such that the restriction of the valuation* *υ*_{p} *to L is the unique extension of v*_{p} to L. Let $\stackrel{~}{L}$ *be the completion of L with respect to the pseudovaluation w*. *Then* $\stackrel{~}{L}$ *is a field*.

*Proof*. We prove that any nonzero element of the closed subring $\stackrel{~}{L}$ of ${\stackrel{~}{\overline{Q}}}_{p}$ is invertible in $\stackrel{~}{L}$. Let *y* ∈ $\stackrel{~}{L}$, *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} ∈ ${C}_{p}^{\sigma}$ C_{p}. In fact *y*_{σ} ∈ $\widehat{L}$ ⊆ **C**_{p} where $\widehat{L}$ denotes the completion of *L* with respect to any of the valuations *σ**υ*_{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 ${\left({\alpha}_{n}^{-1}\right)}_{n}$ is Cauchy with respect to *w* in the complete space $\stackrel{~}{L}$. Thus it has a limit which is also the inverse of *y* in $\stackrel{~}{L}$. □

A converse of the previous proposition is also true.

*Let R* ⊆ ${\stackrel{~}{\overline{Q}}}_{p}$ *be a closed subfield. Then the p-adic valuation υ*_{p} on **Q** *has 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. □

We will make use of the following

*Let x* = (*x*_{σ})_{σ∈G} ∈ ${\stackrel{~}{\overline{Q}}}_{p}$ *satisfying w*(*x*) ≥ 0*. Then there exists q* ∈ **N** {*0*} *such that the sequence x; x*^{q}; :::; x^{qn}; ::: converges in ${\stackrel{~}{\overline{Q}}}_{p}$ *with respect to w and denoting its limit by y* = (*y*_{σ})_{σ∈G} *we have*:
$${\overline{v}}_{p}\left({x}_{\sigma}\right)>0\Rightarrow {y}_{\sigma}=0$$

*and*
$${\overline{v}}_{p}\left({x}_{\sigma}\right)>0\Rightarrow {y}_{\sigma}\ne 0$$

*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 *n*_{0} such that for all *n* ≥ *n*_{0} 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*_{n}_{0} the normal closure of **Q**(*x*_{n}_{0} ). We notice that these images belong to the residue field of the restriction of *υ* to *K*_{n}_{0} . This residue field is a finite field of the form **F**_{q0} for a *p*-power *q*o. Now for each *σ* ∈ *G* we have either *υ*(*x*_{σ}) *>* 0 thus *υ*(${x}_{\sigma}^{{q}_{0}}$) 0 or *υ*(*x*_{σ}) = *0* = *w*(*x*) thus *υ*(${x}_{\sigma}^{{q}_{0}}$) = 0 but *υ*(${x}_{\sigma}^{{q}_{0}}$ - *x*_{σ} > 0. Thus *υ*(${x}_{\sigma}^{{q}_{0}}$ - *x*_{σ} > 0 for all *σ* ∈ *G*. It is known that the function ϕ_{x}: *G* ↦ **R ∪**{∞} defined Φ_{x}(*σ*) = *υ* (${x}_{\sigma}^{{q}_{0}}$ – *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(*x*${x}_{\sigma}^{{q}_{0}}$ – *x*_{σ})≥ *M* for all *σ* ∈ *G*. Since *υ*(*q*o) ≥ 1, it follows that for all *σ* ∈ *G* we have that ${x}_{\sigma}^{{q}_{0}}$ = *x*_{σ} + *z*_{σ} with *z*_{σ} ∈ **C**_{p}, *υ* ≥ *M* and we also have
$$\overline{v}\left({x}_{\sigma}^{{q}_{0}^{2}}-{x}_{\sigma}^{{q}_{0}}\right)=\overline{v}({\left({x}_{\sigma}+{z}_{\sigma}\right)}^{{q}_{o}}-{x}_{\sigma}^{{q}_{0}}=\overline{v}\left({q}_{0}{x}_{\sigma}^{{q}_{0}-1}{z}_{\sigma}+\mathrm{...}+{z}_{{}^{\sigma}}^{{q}_{0}}\right)\ge min\left(1+M,\text{\hspace{0.17em}\hspace{0.17em}}{q}_{0}M\right)$$

and 1 + *M* ∈(1, 2). By replacing *q*o by one of its powers *q* large enough such that *qM >*2 we obtain as above that *υ*(${x}_{\sigma}^{{q}^{2}}-{x}_{\sigma}^{q}$) ≥ 1 + *M* >2 and inductively we obtain that *υ*(${x}_{\sigma}^{{q}^{s}}-{x}_{\sigma}^{{q}^{\left(s-1\right)}}$) > *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.

*Let R be a closed subfield of* ${\stackrel{~}{\overline{Q}}}_{p}$*. Let x* = (*x*_{σ})_{σ∈G} ∈ *R. Then*
$$w\left(x\right)=\overline{v}\left({x}_{\sigma}\right),\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\forall \sigma \in G$$

*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*) = $\frac{r}{t}$ *> 0* with *r; t* positive integers then *w*(*x*^{t}_{)} = *r* thus *w*(*x*^{t}p^{–r}) = 0. Put z = *x*^{t}*p*^{–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 *σ*_{0} ∈ *G* such that *υ*(*x*_{σ}_{0}) > 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*_{σ}_{0} = 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

*Let R be a closed ( with respect to w) subring of* ${\stackrel{~}{\overline{Q}}}_{p}$*, which contains all the negative powers of p, and for which* *υ*(*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* + *y*^{2} + ::: + *y*^{n} + :::) = 1 it follows that *x* is invertible in *R*. We used: *R* is closed and 1 + *y* + *y*^{2} + ::: + *y*^{n} + ::: 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 *υ*for all *α* ∈ *G*. Let *n*_{o} ∈ **N** such that w(*x* – *α*_{n}) *>* 0 for all *n* ≥ *n*_{0}. 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**_{qn0} and *α*_{n} ≠ 0. Thus it follows that ${\alpha}_{n}^{-{q}_{{n}_{0}}-1}$ = 1 that is *υ*_{p}(${\alpha}_{n}^{-{q}_{{n}_{0}}-1}$ –1) *>* 0 thus, by making *n* →1, we obtain *υ*_{p}$({x}_{e}^{{q}_{{n}_{0}}-1}-1)$ *>* 0. It follows that *υ*($({x}_{\sigma}^{{q}_{{n}_{0}}-1}-1)$ –1) > 0 for all *α* ∈ *G* thus *w*(*x*^{qn0–1} – 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*) = $\frac{a}{b}$. Then *w*(*x*^{b}*p*^{a}) = *0* and *x*^{b}*p*^{a} ∈ *R* since *p*^{–a} ∈ *R*. By applying the second case we obtain that *x*^{b}p^{a} is invertible in *R* thus *x* is invertible in *R*.

The statements of the following theorem are straightforward. □

*Let R be a closed subfield of* ${\stackrel{~}{\overline{Q}}}_{p}$*. For each σ* ∈ *G let us define f*_{σ} : *R* → **C**_{p}*, f*_{σ}(*x*) = *x*_{σ}. Then f_{σ} is a field homomorphism satisfying *υ* 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 of ${\stackrel{~}{\overline{Q}}}_{p}$ *to the set of the closed subfields of* **C**_{p}.

${\stackrel{~}{\overline{Q}}}_{p}$ *contains a maximal closed subfield R isomorphic (both algebraically and topologically) to* **C**_{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 $\stackrel{~}{L}$ of *L* with respect to *w* is a subfield of ${\stackrel{~}{\overline{Q}}}_{p}$ and $\stackrel{~}{L}$ is isomorphic to **C**_{p}. Therefore, by Theorem5, it is a maximal subfield of ${\stackrel{~}{\overline{Q}}}_{p}$. □

Let *R* be a closed subfield of ${\stackrel{~}{\overline{Q}}}_{p}$. By applying Theorem 1 of [2] it follows the following theorem.

*Let us denote K*_{e} = *f*_{e}(*R*) *and let z*_{e} ∈ **C**_{p} *such that K*_{e} = $\stackrel{~}{{\mathbf{Q}}_{p}\left[{z}_{e}\right]}$ *as in Theorem1 of [5]. Let* L *be as in the proof of the above corollary and let* (*α*_{n})_{n} *be a sequence in L such that z*_{e} = *lim*_{n}α_{n} with respect to *υ**. For each σ* ∈ Gal(**Q**/Q) *we denote z*_{σ} = *lim*_{n}σ(*α*_{n}) *and K*_{σ} = *f*_{σ}(*K*) *and we have that $\stackrel{~}{{\mathbf{Q}}_{p}\left[{z}_{\sigma}\right]}$. Let z* = (*z*_{σ})_{σ∈G}. *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.

*Let* ${Q}_{p}^{alg}$ = *K be the topological closure of* **Q** *in* **Q** *with respect to* *υ* N*, a fixed extension of υ*_{p} to **Q** *and let R be the maximal subfield of* ${\stackrel{~}{\overline{Q}}}_{p}$ *as in the previous Corollary. Then the subring* _{$\stackrel{~}{K}$} *and ***R** generated by $\stackrel{~}{K}$ *and R is dense in* (${\stackrel{~}{\overline{Q}}}_{p}$$\stackrel{~}{w}$).

*Proof*. Let *R* = $\stackrel{~}{L}$, 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 $\stackrel{~}{K}$$\stackrel{~}{L}$ contains *Q* therefore it is dense in (${\stackrel{~}{\overline{Q}}}_{p}$$\stackrel{~}{w}$). □

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