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Math Help - how to prove this countability

  1. #1
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    how to prove this countability

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  2. #2
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    Show \mathbb{Z}[X_1,X_2,...,X_d] is countable.
    Since by definition, \mathbb{Z}[X_1,...,X_d] = \mathbb{Z}[X_1,...,X_{d-1}][X_d] it remains to prove that if A is countable then A[X] is countable.

    Let A_n[X] = \{ f(x) \in A[x] : \deg f(x) = n \} for n\geq 0.

    Thus, A[x] = \bigcup_{n=0}^{\infty} A_n[x].
    Also, |A_n[x]| =  |A^n| is clear by defining a bijection.
    Since each ( n\geq 0) A^n is countable.
    We have that A[x] is a countable union of countable sets.
    Thus, it is countable.
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  3. #3
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    Show there are uncountably many transcendental numbers
    .

    Given an algebraic number \alpha it needs to satisfy a non-zero polynomial over \mathbb{Q}. There are countably many non-zero polynomials over \mathbb{Q} and each one has finitely many zeros. Since every algebraic number is among those polynomials and conversely a zero of one of those polynomials is algebraic it follows there are countably many algebraic numbers. Thus, there have to be uncountably many transcendental numbers. Because |R| > |N|.
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