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Unique factorization domain, roots of a polynomial, abstract algebra

  1. Sep 9, 2012 #1
    1. The problem statement, all variables and given/known data
    let A be a UFD and K its field of fractions. and [tex]f\in A[x][/tex] where [tex]f(x)=x^{n}+a_{n-1}x^{n-1}+....+a_{1}x+a_{0}[/tex] is a monic polynomial. Prove that if f has a root [tex]\alpha=\frac{c}{d}\in K[/tex],[tex]K=Frac(A)[/tex] then in fact [tex]\alpha\in A[/tex]

    I need some guidance with the proof.
    Proof:
    [tex]f(\alpha)=0\Rightarrow c^{n}+a_{n-1}c^{n-1}d+...+a_{1}cd^{n-1}+a_{0}d^{n}=0[/tex]
    which gives
    [tex]c^{n}=-a_{n-1}c^{n-1}d-...-a_{1}cd^{n-1}-a_{0}d^{n}[/tex]
    and we observe that [tex]d|c^{n}[/tex]*since all the terms on the rhs are multiples of d
    but how do we know that [tex]c^{n}|d[/tex]??? from what do we conclude that
    And later in the proof it says '' hence c,d are relatively prime we get that [tex]\frac{c}{d}\in A[/tex] how do we deduce the last part as well?
    Any help appreciated!

    1. The problem statement, all variables and given/known data
    how do we know that [tex]c^{n}|d[/tex]

    2. The problem statement, all variables and given/known data
    how do we know that c, d are relatively prime?

    3. The problem statement, all variables and given/known data
    based on what do we conclude that [tex]\frac{c}{d}\in A[/tex]
     
  2. jcsd
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