basis of the p-adics $\mathbb{Q}_p$ as a $\mathbb{Q}$-vector space

Question

The p-adics $\mathbb{Q}_p$ are uncountable (because they can be represented by infinite strings of integers in $[0,p-1]$) and hence must be infinite dimensional as a vector space over $\mathbb{Q}$. What is a concrete example of a basis of $\mathbb{Q}_p$ over $\mathbb{Q}$?

Answer 1

There is no concrete example for the same reason that there is no concrete example for a Hamel basis for $\Bbb R$ over $\Bbb Q$.

It is consistent with the failure of the axiom of choice that every linear functional from $\Bbb Q_p$ to $\Bbb Q$ is continuous, and therefore is completely decided by values on a countable set (since $\Bbb Q_p$ is separable). There can only be $2^{\aleph_0}$ of these functionals.

But if there is a basis its cardinality must be $2^{\aleph_0}$ and therefore we can generate $2^{2^{\aleph_0}}$ linear functionals. So in models where the above happens, there is no basis for $\Bbb Q_p$ over $\Bbb Q$. And this means that we can't write an explicit basis, and that we must rely on the axiom of choice for proving it exists.