Is $[\mathbb{R}:\mathbb{Q}]$ countable?

Question

Is there a countable basis for $\mathbb{R}$ as v.s. over $\mathbb{Q}$?

If $K$ is countable field and $S$ is a countable set. Is $K(S)$ countable?

I could only prove if $K$ is countable and $S$ finite, then $K(S)$ countable by induction in $|S|$.

Answer 1

Let $B$ be a linearly independent subset of $\Bbb R$ considered as a vector space over $\Bbb Q$, and suppose that $B$ is countable. Then

$$\operatorname{span}B=\left\{\sum_{b\in F}q_bb:F\subseteq B\text{ is finite and }q_b\in\Bbb Q\text{ for each }b\in F\right\}\;.$$

Since $B$ is countable, it has only countably many finite subsets $F$, and for each choice of $F$ there are only countably many ways to choose the coefficients $q_b$ for $b\in F$, so $\operatorname{span}B$ is countable. Thus, it cannot be all of $\Bbb R$, and it follows that any base for $\Bbb R$ over $\Bbb Q$ must be uncountable.

Answer 2

For each $n$, let $P_n=\{p(s_1,...,s_n):s_1,...,s_n\in S, p\in K[X_1,...,X_n]\}$, then $K(S)=\bigcup_{n=0}^{\infty}P_n$, so if we can show that each $P_n$ is countable, then $K(S)$ is a countable union of countable sets. By induction we can show $K[X_1,...,X_n]$ is countable and the map $f:K[X_1,...,X_n]\times S^n\to P_n:(p(X_1,...,X_n),s_1,...,s_n)\mapsto p(s_1,...,s_n)$ is surjective, so $|P_n|\le |K[X_1,..,X_n]\times S^n|\le \aleph_0$.