Infinite field extension but algebraic subset Is there a subset $ S \subseteq \mathbb{C}$ so that all $s \in S$ are algebraic over $\mathbb{Q}$ and $\left[\mathbb{Q}[S] \colon \mathbb{Q} \right]=\infty$?
$\mathbb{Q}[S]$ is defined as $\mathbb{Q}[S]=\left\lbrace f(x_1,\dotsc,x_n); n \in \mathbb{N}, f \in \mathbb{Q}[X_1,\dotsc,X_n], x_1,\dotsc,x_n \in S \right\rbrace$.

$S$ could be for instance:


*

*all algebraic numbers

*all roots of unity $e^{2\pi i\Bbb Q}$

*all $\sin$'s or $\cos$'s of $2\pi \Bbb Q$

*all $\sqrt[n]{a}$ for $a\in\Bbb Z$ and $n$ fixed

*all $\sqrt[n]{a}$ for $n\in\Bbb Z$ and $a$ fixed

*....

