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 May 19th, 2010, 11:28 AM #1 Senior Member   Joined: Nov 2009 Posts: 129 Thanks: 0 factor ring 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 . <= if g(x) in then f(x) divides g(x). we have that F[x]/ and since g(x) in and f(x) is in F[x] we have that f(x)/g(x). => if f(x) divides g(X) then g(x) in . since f(x)/g(x) in F[x] and we have that F[x]/ hence g(x) is in f(x). May 19th, 2010, 04:54 PM #2 Senior Member   Joined: Apr 2008 Posts: 435 Thanks: 0 Re: factor ring Whoa. What are you doing? This question looks like the definition of an ideal generated by f(x). May 19th, 2010, 05:23 PM #3 Senior Member   Joined: Nov 2009 Posts: 129 Thanks: 0 Re: factor ring <=> if f(x) divides g(x) then g(x) in Proof: Suppose f(x) divides g(x)q(x). then g(x)q(x) in . which is maximal. Therefore is a prime ideal. Hence g(x)q(x) in . implies that either g(x) in giving f(x) divides g(x) or that q(x) in giving f(x) divides q(x). But we want that g(x) in giving f(x) divides g(x). can this prove go both way if it is right? May 21st, 2010, 04:39 AM #4 Senior Member   Joined: Apr 2008 Posts: 435 Thanks: 0 Re: factor ring Ah, those are supposed to mean 'divides.' You see, when I see A/B - I see A quotiented out by the ideal of B. When I see A|B, I see that A divides B. This is good, as I didn't understand the first post at all. Tags factor, ring Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Mathew Abstract Algebra 5 August 29th, 2010 08:53 PM tinynerdi Abstract Algebra 5 May 16th, 2010 11:42 AM cgouttebroze Abstract Algebra 5 August 14th, 2008 12:04 PM stf123 Abstract Algebra 3 December 7th, 2007 07:47 AM Frazier001 Abstract Algebra 1 December 6th, 2007 01:21 PM

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