Results 1 to 2 of 2
Like Tree1Thanks
  • 1 Post By johng

Math Help - UFD? Nature of polynomial x in the Ring Q_Z[x]

  1. #1
    Super Member Bernhard's Avatar
    Joined
    Jan 2010
    From
    Hobart, Tasmania, Australia
    Posts
    558
    Thanks
    2

    UFD? Nature of polynomial x in the Ring Q_Z[x]

    Let  \mathbb{Q}_\mathbb{Z}[x] denote the set of polynomials with rational coefficients and integer constant terms.

    (a) Show that the only divisors of x in  \mathbb{Q}_\mathbb{Z}[x] are the integers (constant polynomials) and first degree polynomials of the form  \frac{1}{n}x with  0 \ne n \in \mathbb{Z}

    (b) For each non-zero  n \in \mathbb{Z} show that the polynomial  \frac{1}{n}x is not irreducible in  \mathbb{Q}_\mathbb{Z}[x]

    (c) Show that x cannot be written as a finite product of irreducible elements in  \mathbb{Q}_\mathbb{Z}[x]

    Would appreciate help with this exercise

    Peter
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Dec 2012
    From
    Athens, OH, USA
    Posts
    639
    Thanks
    257

    Re: UFD? Nature of polynomial x in the Ring Q_Z[x]

    Hi again,
    This continues the problem of your previous posting. What this does is prove the ring in question is not a unique factorization domain. Here's a solution:

    UFD? Nature of polynomial x in the Ring Q_Z[x]-mhfrings4.png
    Thanks from Bernhard
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Zeros of a polynomial in a ring.
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: June 25th 2013, 03:28 AM
  2. Polynomial Ring Z[x]/(x^2)
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: April 4th 2013, 09:22 PM
  3. Replies: 1
    Last Post: October 23rd 2011, 06:36 AM
  4. Polynomial ring
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: February 2nd 2011, 01:30 PM
  5. polynomial ring
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: October 24th 2009, 03:13 AM

Search Tags


/mathhelpforum @mathhelpforum