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Factor ring

  1. May 19, 2010 #1
    1. The problem statement, all variables and given/known data
    let F be a field and f(x),g(x) in F[x]. Show that f(x) divides g(x) if and only if g(x) in <f(x)>

    2. Relevant equations
    let E be the field F[x]/<f(x)>

    3. The attempt at a solution
    <=> if f(x) divides g(x) then g(x) in <f(x)>
    Proof: Suppose f(x) divides g(x)q(x). then g(x)q(x) in <f(x)>. which is maximal. Therefore <f(x)> is a prime ideal. Hence g(x)q(x) in <f(x)>. implies that either g(x) in <f(x)> giving f(x) divides g(x) or that q(x) in <f(x)> giving f(x) divides q(x). But we want that g(x) in <f(x)> giving f(x) divides g(x).

    can this prove go both way if it is right?
  2. jcsd
  3. May 20, 2010 #2
    I think this is even simpler than you think. The condition for g(x) to divide f(x) is that there is q(x) in F[x] such that f(x)=q(x)g(x), and this is exactly the condition for g(x) to belong to the ideal generated by f(x).
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