Simple extension of a countable field is also countable?

Question

How do we know "Simple extension of a countable field is also countable?" This intuitively makes sense but I'm not sure how to rigorously prove this. I want to use this to prove that there does not exist a simple extension from $\mathbb{Q}$ to $\mathbb{R}$. Since $\mathbb{Q}$ is countable, and any simple extension from $\mathbb{Q}$ is countable, but $\mathbb{R}$ is uncountable so there does not exist such extension...

Answer 1

If you can show that the field $K(x)$ is countable whenever $K$ is, then that should give you enough to finish solving the more general problem. To this end, complete two steps:

• If $K$ is countable then the polynomial ring $K[x]$ is countable. This is a simple application of the fact that the set of polynomials of degree $n$ is clearly countable, and the union of such sets is the entire ring.

• If a ring is countable, so is its field of fractions. To prove this, follow the ideas used in proving that the field of fractions of $\mathbb{Z}$ is countable.