My Math Forum  

Go Back   My Math Forum > College Math Forum > Abstract Algebra

Abstract Algebra Abstract Algebra Math Forum


Reply
 
LinkBack Thread Tools Display Modes
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 <f(x)>.

<= if g(x) in <f(x)> then f(x) divides g(x).
we have that F[x]/<f(x)> and since g(x) in <f(x)> and f(x) is in F[x] we have that f(x)/g(x).

=> if f(x) divides g(X) then g(x) in <f(x)>.
since f(x)/g(x) in F[x] and we have that F[x]/<f(x)> hence g(x) is in f(x).
tinynerdi is offline  
 
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).
jason.spade is offline  
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 <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?
tinynerdi is offline  
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.
jason.spade is offline  
Reply

  My Math Forum > College Math Forum > Abstract Algebra

Tags
factor, ring



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Ring Mathew Abstract Algebra 5 August 29th, 2010 08:53 PM
factor ring of a field tinynerdi Abstract Algebra 5 May 16th, 2010 11:42 AM
Ring and pseudo-ring cgouttebroze Abstract Algebra 5 August 14th, 2008 12:04 PM
ring stf123 Abstract Algebra 3 December 7th, 2007 07:47 AM
ring Frazier001 Abstract Algebra 1 December 6th, 2007 01:21 PM





Copyright © 2019 My Math Forum. All rights reserved.