1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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).
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Factor ring
  1. Factor ring of a ring (Replies: 3)

Loading...