Prove that tr.deg($\displaystyle \mathbb{C}$/$\displaystyle \mathbb{Q}$)=$\displaystyle \infty$.

What is the cardinality of the transcendence base?

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- Apr 21st 2011, 11:39 PMKaKatranscendence degree
Prove that tr.deg($\displaystyle \mathbb{C}$/$\displaystyle \mathbb{Q}$)=$\displaystyle \infty$.

What is the cardinality of the transcendence base? - Apr 22nd 2011, 02:27 PMHappyJoe
What are your thoughts thus far?

This might get you going: Recall that the set of complex numbers that are algebraic over Q is countable. Hence there exists a complex number z_1, which is not algebraic over Q. However, there are still only a countable number of elements in C that are algebraic over Q(z_1).

You also need to recall that a countable union of countable sets is itself a countable set.

Feel free to ask, if you have specific questions to my response.