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.
> 
> 
> 



p-adic expansion for elements in algebraic closure of p-adic numbers
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.


