In the following I will describe a proposal for the p-adic expansion of the elements of the algebraic closure $\overline{\mathbb{Q}_p}$ of $\mathbb{Q}_p$. My question is if this "conjecture" has been proved or disproved before.

Consider the following result proved in "Local Fields" Berlin. (1980) by Serre, J. P.:

Theorem: If $R$ is a perfect field of characteristic $p>0$, then there exists a unique ring $W(R)$ with characteristic $0$ with discrete valuation $v$ such that the residue field is $R$, $v(p)=1\in\mathbb{Z}$, and $W(R)$ is complete with respect to $v$. Also, the field of fractions $F(W(R))$ is the only field with characteristic $0$ with discrete valuation $v$ such that the residue field is $R$, $v(p)=1\in\mathbb{Z}$, and $F(W(R))$ is complete with respect to $v$.

In this context, the valuation ring of $F(W(R))$ is $W(R)$ the ring of Witt vectors with coefficients in $R$. For example, if $R=\mathbb{F}_p$, then $W(R)=\mathbb{Z}_p$ and $F(W(R))=\mathbb{Q}_p$.

In the article "Maximally complete fields." Enseign. Math.(2) 39.1-2 (1993): 87-106, @Bjorn Poonen describes the unique maximally complete immediate extension field of $\overline{\mathbb{Q}_p}$ as follows:

let $W(\overline{\mathbb{F}_p})((t^\mathbb{Q}))$ be the ring of all series $\sum_{g}\alpha_gt^g$ where $g\in\mathbb{Q}$, $\alpha_g\in W(\overline{\mathbb{F}_p})$, and $\{g:\alpha_g\neq0\}$ is well-ordered. A series $\sum_{g}\alpha_gt^g\in W(\overline{\mathbb{F}_p})((t^\mathbb{Q}))$ is null if for all $g\in\mathbb{Q}$, $\sum_{n\in\mathbb{Z}}\alpha_{g+n}p^n=0$ in $F(W(\overline{\mathbb{F}_p})).$ The set $N$ of all null series is an ideal of $W(\overline{\mathbb{F}_p})((t^\mathbb{Q}))$ that contains the polynomial $t-p$. The quotient $$L=W(\overline{\mathbb{F}_p})((t^\mathbb{Q}))/N$$ is a field of characteristic $0$, valuation ring $\mathbb{Q}$ and residue class field $\overline{\mathbb{F}_p}$. Also $L$ is algebraically closed, and it is the maximally complete immediate extension of $\overline{\mathbb{Q}_p}$. Furthermore, if $S\subset W(\overline{\mathbb{F}_p})$ is a set of representatives of the classes of $\overline{\mathbb{F}_p}$, then each series $\sum_{g}\alpha_gt^g$ in $W(\overline{\mathbb{F}_p})((t^\mathbb{Q}))$ is null equivalent to a unique series of the form $\sum_{g}\beta_gt^g$ for $\beta_g\in S$. In other words, we have the following expansion for the elements of $L$:

$$L=\bigg\{\sum_{g\in\mathbb{Q}}\beta_gp^g:\beta_g\in S, \{g:\beta_g\neq0\}\mbox{ is well-ordered}\bigg\}.$$

Notice the analogy with the $p$-adic expansion for elements of $\mathbb{Q}_p$: $$\mathbb{Q}_p=\mathbb{Z}_p((t^\mathbb{Z}))/N=W(\mathbb{F}_p)((t^\mathbb{Z}))/N=\bigg\{\sum_{n=m}^\infty\beta_np^n:m\in\mathbb{Z},\beta_n=0,1,\dots,p-1 \bigg\}.$$

Now let's consider the following chain of valued field extensions: $$\mathbb{Q}_p((x))\subset \overline{\mathbb{Q}_p}((x))\subset \overline{\overline{\mathbb{Q}_p}((x))}= \bigcup_{n=1}^\infty \overline{\mathbb{Q}_p}((x^\frac{1}{n}))\subset\overline{\mathbb{Q}_p}((t^\mathbb{Q}))$$

Also consider the following chain of rings: $$\mathbb{Z}_p((x))\subset W(\overline{\mathbb{F}_p})((x))\subset \bigcup_{n=1}^\infty W(\overline{\mathbb{F}_p})((x^\frac{1}{n}))\subset W(\overline{\mathbb{F}_p})((t^\mathbb{Q}))$$

By taking the quotient of these rings by their ideals of null series, we obtain the following chain of field extensions: $$\mathbb{Q}_p= \mathbb{Z}_p((x))/N\subset W(\overline{\mathbb{F}_p})((x))/N\subset \bigcup_{n=1}^\infty W(\overline{\mathbb{F}_p})((x^\frac{1}{n}))/N\subset W(\overline{\mathbb{F}_p})((t^\mathbb{Q}))/N=L$$ where there is an abuse of notation for the ideals $N$ (The N's of different quotients are different).

Conjecture 1: The relation
$$\overline{\overline{\mathbb{Q}_p}((x))}= \bigcup_{n=1}^\infty \overline{\mathbb{Q}_p}((x^\frac{1}{n}))$$ implies the relation $$\overline{\mathbb{Q}_p}= \bigcup_{n=1}^\infty W(\overline{\mathbb{F}_p})((x^\frac{1}{n}))/N= \bigg\{\sum_{n=m}^\infty\beta_np^\frac{n}{N}:m,N\in\mathbb{Z},N\geq1,\beta_n\in S \bigg\},$$ where $S\subset W(\overline{\mathbb{F}_p})$ is a set of representatives of the classes of $\overline{\mathbb{F}_p}$.

Edit: As it is noticed in the comments below, the conjecture 1 is false, since $\sqrt{-1}\in\overline{\mathbb{Q}_2}$ cannot be represented in the proposed form. Also notice that $W(\overline{\mathbb{F}_p})\neq\{x\in\overline{\mathbb{Q}_p}:|x|_p\leq1\}$ since the former has a discrete valuation while latter has a dense valuation. Also I just noticed the following result, which follows immediately by the theorem stated above and the properties of the maximally complete field $L$ described by Bjorn Poonen in its paper also mentioned above. The notations are defined above.

Proposition: If $R$ is a perfect field of characteristic $p>0$, then $W(R)((x))/N$ is the only field of characteristic $0$, with discrete valuation $v$ such that the residue class field is $R$, $v(p)=1\in\mathbb{Z},$ and $W(R)((x))/N$ is complete with respect $v$. Hence $F(W(R))=W(R)((x))/N$ is maximally complete.

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    $\begingroup$ In $\overline{\mathbb{Q}_2}, \sqrt{-1} = 1+2^{1/2}+2^{3/4}+2^{7/8}+...+\zeta_3\times2+...$. $\endgroup$ – David Lampert Dec 1 '16 at 23:47
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    $\begingroup$ It's not true that $W(\bar{\mathbb F}_p) = \mathcal O_{\bar{\mathbb Q}_p}$. You only get the maximal unramified extension of $\mathbb Q_p$. $\endgroup$ – R. van Dobben de Bruyn Dec 1 '16 at 23:50

The following article discusses p-adic expansions in $\overline{\mathbb{Q}}_p$ and of $\mathbb{C}_p$.

Algebraic $p$-adic expansions, David Lampert, Journal of Number Theory 23 (1986), 279–284.

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    $\begingroup$ Thanks for the article. When I was reading it and I found the notation $(1/Np^\infty)\mathbb{Z}$ and I can't deduce its meaning. Do you know it? $\endgroup$ – Chilote Dec 2 '16 at 5:31
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    $\begingroup$ It means $\{m/{Np^k}: m \in \mathbb{Z}, k \in \mathbb{N}\}$. $\endgroup$ – David Lampert Dec 2 '16 at 13:26
  • $\begingroup$ @DavidLampert I just wanted to notice that in Theorem 1 of your paper "Algebraic $p$-adic expansions" the exponents $r_i$ can be written explicitly as $r_i=\frac{1}{p-1}-\frac{1}{p^i}$ for all $i\in\mathbb{N}$. $\endgroup$ – Chilote Dec 14 '16 at 18:47
  • $\begingroup$ @Chilote Thanks for your comment. I think there's a mistake in the Theorem 1 description of the exponents in the expansion of $1^{1/p^2}$, according to my notes that I wrote years after the article was published. The expansion should be $1^{1/p^2} = 1 - \zeta_1{p}^{1/p(p-1)} - \zeta_2{p}^{2/p(p-1)} - ... - \zeta_{p-2}{p}^{(p-2)/p(p-1)}$ (the same $\zeta_i$ as in $1^{1/p}$), and the first accumulation value of the exponents is $1/(p-1)$ as in the article. For example (according to my notes), $1^{1/9} = 1 + \zeta_4{3^{1/6}} + 3^{1/3} - 3^{4/9} + 3^{13/27} - 3^{40/81} ...$ $\endgroup$ – David Lampert Dec 14 '16 at 22:42

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