The wikipedia article on p-adic numbers warns about $b$-adic expansions where $b$ is not a prime:
Although for p-adic numbers p should be a prime, base 10 was chosen to highlight the analogy with decimals. The decadic numbers are generally not used in mathematics: since 10 is not prime, the decadics are not a field.
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As noted above, 10-adic numbers have a major drawback. It is possible to find pairs of non-zero 10-adic numbers (having an infinite number of digits, and thus not rational) whose product is 0.
Hence I was quite surprised to learn that quote notation, which "follows the approach of Kurt Hensel's p-adic numbers", also works if $b$ is not a prime. So I wonder whether anything will go wrong if also non-terminating expansions which are not periodic will be allowed.
Let $b=p_1^{\alpha_1}\dots p_k^{\alpha_k}$ with prime numbers $p_i$ and positive integers $\alpha_i$. My guess is that any number $x \in\mathbb Q_{p_1} \cap\dots\cap \mathbb Q_{p_k}$ can be represented by a potentially non-terminating $b$-adic expansions, and that every non-terminating $b$-adic expansions is equivalent to such a number. You may object that this "formal intersection" is meaningless, but because $\mathbb Q_p$ can be represented by equivalence classes of certain sequences of rational numbers, it should be possible to compute this intersection in the space of sequences of rational numbers.
How could such a $b$-adic expansions be defined? My guess is that $x=p_1^{\beta_1}\dots p_k^{\beta_k}\sum_{i=0}^\infty d_i b^i$ with (potentially negative) integers $\beta_i$, integers $d_i$ with $0\leq d_i < b$ and $\operatorname{gcd}(d_0,b)=1$. Note that using $p_1^{\beta_1}\dots p_k^{\beta_k}$ instead of $b^\beta$ (which allows for $\operatorname{gcd}(d_0,b)=1$) avoids the counterexample given in the wikipedia article. One obvious question is whether addition, subtraction, multiplication and division will be well defined for this representation. This seems to be easy to check, at least for addition, subtraction and multiplication. But is there anything else which needs to be checked, before one can conclude that this is a valid number representation for the field $\mathbb Q_{p_1} \cap\dots\cap \mathbb Q_{p_k}$?
Best Answer
$\mathbb Q_p$ is the completion of $\mathbb Q$ under the $p$-adic metric. With several norms $|\cdot|_{p-1}, \ldots, |\cdot|_{p_k}$, you can combine these to a single norms e.g. by adding: $|x|:=|x|_{p_1}+\ldots+|x|_{p_k}$, and consider the completion with respect to this norm.