ac.commutative algebra - What is the transcendence degree of Q_p and C over Q?

In both cases the transcendence degree is the cardinality of the continuum. CH is not needed.

This is a corollary of the following result: let $K$ be any infinite field, and let $L/K$ be any extension. Then

$# L = operatorname{max} (# K, operatorname{trdeg}_K L)$.

To prove this, in turn it suffices to establish the following two results (each of which is straightforward):

1) If $K$ is infinite and $L/K$ is algebraic, then $# L = # K$.

2) If $K$ is any infinite field, $T = {t_i}_{i in I}$ is an arbitrary set of indeterminates and $K(T)$ is a purely transcendental function field in the indeterminates $T$, then $ # K(T) leq # T + # K$.