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Countability of the rationals

Does the following identity holds :
$$\mathbb{Q} \text{ is countable } \quad\Leftrightarrow\quad \mathbb{Q} \text{ is a Dedekind dense set in } (\mathbb{R},+,\times)$$
Could someone provide a good reference for the proof of this fact?

A:

No, this is false. The countability of $\mathbb{Q}$ is

essentially equivalent to the hereditariness of $\mathbb{Q}$ as a field
extension of $\mathbb{Q}$ by $\mathbb{R}$.

The countability of $\mathbb{Q}$ is not equivalent to the hereditariness of $\mathbb{Q}$ as a field extension of $\mathbb{Q}$ by $\mathbb{R}$. We can take $\mathbb{Q}$ to be a countable field extension of $\mathbb{Q}$ by $\mathbb{R}$ where the quotient field of $\mathbb{Q}[X]$ is just the rationals.
Now let $K$ be any countable field. Since $K$ is countable, it is closed under scalar multiplication and $K[X]$ is still countable. Therefore $K[X]$ contains a maximal ideal $M$, i.e. $M$ is a maximal ideal of $K[X]$. So $1\in M$ and there exists a prime element $p\in K$ such that $p otin M$. Thus $p$ is a prime element of $\mathbb{Q}$, but this field is countable, so we have a contradiction.

A:

$$\mathbb{Q}\quad \text{is countable}\tag{1}$$
$\Leftrightarrow \quad \{p\in \mathbb{R}\mid p\in \mathbb{Q}\}$ is countable in $\mathbb{R}$
\$\Rightarrow

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