What's a p-adic representative map?

Question

For the sake of a simple example I can work with, let $\Bbb Q_2[\zeta]$ be some extension of a set of 2-adic numbers to include a square root of two.

I interpret this comment to mean these numbers can be written with radix $\{0,1,2,3\}$ and in general $m^{th}$ roots of $p^n$ with $m<n$ can be represented with the radix $\{0,\ldots p^n-1\}$. But what is the base here? Is it still two, or is it now some square root of two (or in general some $m^{th}$ root of $p^n$?

The term "representative map" appears to be in general usage but I can't find a definition. I'm assuming this is similar to picking Teichmuller representatives, right?

How do I lift the square roots of two to representatives? From my limited superficial experience I'm guessing the process is something like: take a polynomial $P(x)=x^2-2$ whose zeroes are roots of two and keep taking residues mod $p^n$ for increasing $n$. But I'm not clear how this works in base $\sqrt2$ and with radix $\{0,1,2,3\}$

Although my specific query here is in base $2$ I think the general question is What's a representative map?

Answer 1

In my comment you link to, all I am saying is that if $x \in \mathbb Z_p$, then an expression like

$$ x- (x \text{ mod } p^n)$$

(which you write in your question there) does not make sense, because $x \text{ mod } p^n$ is an element of the quotient $\mathbb Z_p / (p^n)$, and you cannot add or subtract elements of $\mathbb Z_p / (p^n)$ to or from elements of $\mathbb Z_p$.

From the context I guessed what you actually meant there was the truncation map

$$\sum_{k \ge 0} a_k p^k \mapsto \sum_{k \ge n+1} a_k p^k$$

where I further guessed that you use the popular although non-canonical choice of the $a_k$ being chosen from the natural numbers $\{0, \dots, p-1\}$. Which are representatives of the elements of the quotient $\mathbb Z_p/(p)$ in $\mathbb Z_p$. More generally, the natural numbers $\{0, \dots, p^n-1\}$ form a set of representatives of $\mathbb Z_p/(p^n)$ in $\mathbb Z_p$, and if

$$[ \cdot ] : \mathbb Z_p/(p^n) \rightarrow \mathbb Z_p$$

denotes the "representative map" which to each element of the quotient $$0 \text{ mod } p^n, 1 \text{ mod } p^n, \dots, p^n-1 \text{ mod } p^n$$ assigns the representative $$0, 1, \dots p^n -1,$$ then you can write the above truncation map conveniently as:

$$x \mapsto x - [x\text{ mod } p^n]$$

This is also convenient in that it shows that your truncation map, maybe unbeknownst to you, depends on that choice of representatives, and any different choice of a representative map would lead to a different "truncation" map which shares some but not all properties with the one you care about.

As a final note, for what it's worth, nowhere in here do I leave the $p$-adic integers. I know you want to adjoin some $\zeta$, but so far this issue here has absolutely nothing to do with that.