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Why does C have transcendence degree c over Q? Jan 28 '12 Mathematics Stack Exchange
7
(Of course I assume the Axiom of Choice...) Choose a transcendence basis X={xi}i∈I for C over Q. Then C is an algebraic extension of Q(X). Now here are two rather straightforward facts:
1: If F is any infinite field and K/F is an algebraic extension, then #K=#F.
2: For any infinite field F and purely transcendental extension F(X), we have #F(X)=max(#F,#X).
Putting these together we find
c=#C=#Q(X)=max(?0,#X).
Since c>?0, we conclude c=#X.