Is the tr.deg of Q\_p over Q 1? and what about C over Q?



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$. 


