Results 1 to 2 of 2

Math Help - Zero Divisors in Polynomial Ring

  1. #1
    Junior Member
    Joined
    Mar 2007
    Posts
    40

    Zero Divisors in Polynomial Ring

    I am looking to prove the following

    Prove that if f(x) is an element of F[x] is not irreducible, then F[x]/f(x) contains zero-divisors.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by chadlyter View Post
    I am looking to prove the following

    Prove that if f(x) is an element of F[x] is not irreducible, then F[x]/f(x) contains zero-divisors.
    Not irreducible over F means reduccible over F (you just phrased it in a very strange way).

    Now, F[x]/<f(x)> is an integral domain if and only if <f(x)> is a prime ideal.

    To complete the proof show that <f(x)> is not a prime ideal (of course you use the fact that f(x) is reducible). We show this like this. <f(x)> is a principal ideal by definition. And f(x)=p(x)q(x) where deg {p(x)} and deg {q(x)} are both strictly less then deg{f(x)} because it is reduccible. Then since f(x) in <f(x)> it means p(x)q(x) in <f(x)> by our assumption that this principal ideal is a prime ideal would imply that p(x) in <f(x)> or q(x) in <f(x)> which is not possible because the degree of any element in <f(x)> is at least as large as f(x) because <f(x)> = {r(x)f(x) | r(x) in F[x]}, and hence cannot be in this ideal. Thus, the assumption that <f(x)> was a prime ideal was a contradiction.
    Q.E.D.


    ---
    What does this have to do with a ring homomorphism?
    Last edited by ThePerfectHacker; April 17th 2007 at 08:39 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: October 23rd 2011, 07:36 AM
  2. A commutative Ring R has no zero divisors
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: August 1st 2011, 08:04 AM
  3. A Ring's zero-divisors
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: September 29th 2010, 02:34 PM
  4. Ring problem with zero divisors
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 8th 2010, 02:46 PM
  5. [SOLVED] the set of zero-divisors of a ring
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: March 21st 2010, 02:21 AM

Search Tags


/mathhelpforum @mathhelpforum