Results 1 to 2 of 2

Math Help - Exercise - Irreducible Polynomial

  1. #1
    Junior Member
    Joined
    Jun 2009
    Posts
    58

    Exclamation Exercise - Irreducible Polynomial

    Prove that an irreducible polynomial in Q[x] has no multiples roots

    Thanks in advance.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member Gamma's Avatar
    Joined
    Dec 2008
    From
    Iowa City, IA
    Posts
    517

    derivative

    Let f(x)\in \mathbb{Q}[x] be irreducible. Then consider the derivative. f'(x) is a polynomial of degree n-1 and since char(\mathbb{Q})=0, so it is not identically 0. Up to constant factors, the only factors of f(x) are f(x) and 1, so f(x) and f'(x) are relatively prime, thus f(x) is separable, ie has no multiple roots.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Irreducible Polynomial
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: January 27th 2011, 07:42 PM
  2. irreducible polynomial
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: June 4th 2010, 05:40 AM
  3. Irreducible polynomial
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: February 9th 2010, 01:08 AM
  4. Another polynomial exercise
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: June 6th 2009, 10:01 PM
  5. how do you know if a polynomial is irreducible
    Posted in the Number Theory Forum
    Replies: 0
    Last Post: February 17th 2009, 04:13 PM

Search Tags


/mathhelpforum @mathhelpforum