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. Wondershare Dr.Fone Toolkit For Iso Pc Mac 10.8.9.86 FULL Crackl com. Wondershare Dr.Fone Toolkit For Iso Pc Mac 10.8.9.86 FULL Crackl.Q:
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
https://colab.research.google.com/drive/1lI8SOVPqjolehpWym9YhfXqQ2zW_3xIx
https://ello.co/7castmeragi/post/xkq7o9vxfzmfp-h17nip1q
https://ello.co/caloqaph-go/post/vz4im_smw4yxcq6dccgg4q
https://ello.co/9enlivlinc_ha/post/egthiagcj9dau6nhsmcy3q
https://ello.co/girbaycrosin/post/dprlefupanyaxnoad4b6eg
https://colab.research.google.com/drive/1KTvtmx6vufwLcgt6wxQY-BvXnbgjho-w
https://ello.co/refnemono/post/fmufzolqkwrf2j_b9nj71w
https://colab.research.google.com/drive/1qHAbJDA2Bc3ocyqLzwKOFwfYvZ1lp6If
https://colab.research.google.com/drive/1Tl1aA7pJ23gUoqTNsjRY1dCPZSeF91-E
https://ello.co/alabocresbo/post/bydoiplgr2yuklxudec_ag
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