Results 1 to 4 of 4

Math Help - Probably a simple explanation

  1. #1
    Junior Member
    Joined
    Sep 2008
    From
    Indiana
    Posts
    26

    Probably a simple explanation

    Could someone explain why 2x-10 is not an irreducible in Z[x] but is an irreducible in Q[x].
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by Ryaη View Post
    Could someone explain why 2x-10 is not an irreducible in Z[x] but is an irreducible in Q[x].
    It factorises as 2(x5) in both cases. However, 2 is a unit in \mathbb{Q}[x] but not in \mathbb{Z}[x].
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Sep 2008
    From
    Indiana
    Posts
    26
    Quote Originally Posted by Opalg View Post
    It factorises as 2(x5) in both cases. However, 2 is a unit in \mathbb{Q}[x] but not in \mathbb{Z}[x].
    Why would that not be the other way around? (since it is irreducible in Q[x] and reducible in Z[x])
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by Ryaη View Post
    Why would that not be the other way around? (since it is irreducible in Q[x] and reducible in Z[x])
    Reducible means having a nontrivial factorisation. Irreducible means having no nontrivial factorisation. Nontrivial means that neither of the factors is a unit. If one of the factors is a unit then the factorisation doesn't count as a nontrivial factorisation. Since 2 is a unit in \mathbb{Q}, the factorisation 2x – 10 = 2(x – 5) doesn't count as nontrivial. That is why 2x – 10 is irreducible in \mathbb{Q}[x].
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. simple explanation about translation vectors
    Posted in the Calculus Forum
    Replies: 5
    Last Post: January 15th 2012, 11:41 PM
  2. ACF() Explanation
    Posted in the Statistics Forum
    Replies: 0
    Last Post: January 18th 2011, 10:24 PM
  3. i need explanation
    Posted in the Geometry Forum
    Replies: 1
    Last Post: July 9th 2008, 09:00 AM
  4. Replies: 5
    Last Post: April 21st 2008, 08:33 PM
  5. A simple metric explanation
    Posted in the Advanced Math Topics Forum
    Replies: 3
    Last Post: June 27th 2006, 01:46 PM

Search Tags


/mathhelpforum @mathhelpforum