dimension of $\mathbb{C}$ over $\mathbb{Q}$

Question

I am studying about extension fields.I want to know what would be [$\mathbb{C}$ : $\mathbb{Q}$] ?

Answer 1

$\mathbb Q$ is countable, while $\mathbb C$ is uncountable. Thus $[\mathbb C:\mathbb Q]$ is inifinite, and in particular, uncountable, of the same cardinality as $\mathbb C$.

Answer 2

Another way to see it is to notice that $\mathbb Q\subseteq\mathbb Q(\pi)\subseteq\mathbb C$, hence $[\mathbb C:\mathbb Q]\geq[\mathbb Q(\pi):\mathbb Q]=\infty$ being $\pi$ trascendental over $\mathbb Q$.