What is the cardinality of a transcendence basis of $\mathbb{C}$ over $\mathbb{Q}$? What is the cardinality of a transcendence basis of $\mathbb{C}$ over $\mathbb{Q}$? Is it that of the continuum? Proof?

If $S$ has infinite cardinality $\kappa$, then $|\mathbb Q(S)|=\kappa$. And if $|F|=\kappa$, then $|F[X]|=\kappa $ and finally $|\bar F|=\kappa$. Therefore we need exactly $\kappa=2^{\aleph_0}$ if we want $\overline{\mathbb Q(S)}=\mathbb C$.