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Countability homework problem

  1. Sep 17, 2009 #1
    1. The problem statement, all variables and given/known data

    A real number [tex]x \in R[/tex] is called algebraic if there exist integers [tex]a_{0},a_{1},a_{2}...,a_{n}[/tex], not all zero, such that

    [tex]a_{n}x^{n} + a_{n}_{1}x^{n-1} + ... + a_{1}x + a_{0} = 0 [/tex]

    Said another way, a real number is algebraic if it is the root of a polynomial with integer coefficients...

    Fix [tex]n \in N[/tex], and let [tex]A_{n}[/tex] be the algebraic numbers obtained as roots of polynomials with integer coefficients that have degree [tex]n[/tex]. Using the fact that every polynomial has a finite number of roots, show that [tex]A_{n}[/tex] is countable. (For each [tex]m \in M[/tex], consider polynomials [tex]a_{n}x^{n} + a_{n}_{1}x^{n-1} + ... + a_{1}x + a_{0} [/tex] that satisfy [tex]\left|a_{n}\right| + \left|a_{n-1}\right| + ... + \left|a_{1}\right| + \left|a_{0}\right| \leq m [/tex].)


    2. Relevant equations



    3. The attempt at a solution

    I'm not sure how to explain this coherently... here is what I have. I feel like there are some holes.

    -Every polynomial has a finite # of roots (therefore it is countable)

    -[tex]m \in N[/tex] is the sum of all integer coefficients for the roots of polynomials.

    -Let [tex]C_{m}[/tex] be a set containing all possible polynomials whose integer coefficients add up to [tex]m[/tex] for a fixed [tex]n[/tex]. Since there are finite ways to express [tex]m[/tex] as a sum of integers, each [tex]C_{m}[/tex] is countable.

    -Every [tex]A_{n}[/tex] is made up of [tex]C_{m}[/tex], so [tex]A_{n}[/tex] is countable (since union of a countable # of countable sets is countable).
     
  2. jcsd
  3. Sep 17, 2009 #2

    Dick

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    Re: Countability

    I don't see anything wrong with that. As you said, a union of countable sets is countable. That's the point, right?
     
  4. Sep 17, 2009 #3
    Re: Countability

    Yeah, what I came up with at the end is probably correct... I guess I'm more concerned about whether each step makes sense or if there's anything I should clarify more.
     
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