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

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

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$.