Showing that $\mathbb Q$ is not complete with respect to the 2-adic and 3-adic absolute value

Question

I have seen here how to show that $\mathbb Q$ is not complete with respect to the $p$-adic absolute value, where $p\geq5$. Is there a similar proof/idea for $p=2$ and $p=3$?

Answer 1

Take a monic polynomial $Q(X) \in \mathbb Z[X]$ that does not have a rational root, but which has a root $a$ modulo $p$, and with $Q'(a) \not \equiv 0 \pmod p$. By Hensel's lemma, $Q(X)$ then has a root in $\mathbb Q_p$ (even in $\mathbb Z_p$), which is necessarily an irrational $p$-adic. Hence $\mathbb Q_p \neq \mathbb Q$, so that $\mathbb Q$ is not $p$-adically complete.

For $p \geq 5$, you can take $Q(X) = \frac{X^{p-1}-1}{X^2-1}$ and you recover more or less the construction in the linked answer.

For $p = 2$, you can take for example $Q(X) = X^3+3$.

For $p = 3$, you can take for example $Q(X) = X^2 + 2$.