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Math Help - transcendence degree

  1. #1
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    transcendence degree

    Prove that tr.deg( \mathbb{C}/ \mathbb{Q})= \infty.
    What is the cardinality of the transcendence base?
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  2. #2
    Member HappyJoe's Avatar
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    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.
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