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Math Help - polynomials

  1. #1
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    polynomials

    how do I show that the set of polynomials with interger coefficients is countable?
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  2. #2
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    See my response to your similar question about the algebraic numbers. If you're still stuck, say where you're stuck and I'll help you with the details.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by ricky View Post
    how do I show that the set of polynomials with interger coefficients is countable?
    The set of all such polynomials of degree less than or equal to n, denoted by \mathbb{Z}_n[x] is clearly countable via the obvious identification \displaystyle f:\mathbb{Z}_n[x]\to\mathbb{Z}^{n+1}:\sum_{k=0}^{n}\alpha_k x^k\mapsto (\alpha_0,\cdots,\alpha_k). Note then that \displaystyle \mathbb{Z}[x]=\bigcup_{n\in\mathbb{N}}\mathbb{Z}_n[x] and so..
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