1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Unit in a ring (abstract algebra)

  1. Feb 26, 2013 #1
    1. The problem statement, all variables and given/known data
    Is (x^2-1) a unit in F[x]? where F is a field.


    2. The attempt at a solution
    I might say yes, cause we can find the taylor expansion of 1/(x^2-1), is my idea right??????
     
  2. jcsd
  3. Feb 26, 2013 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    The taylor expansion is not a polynomial. It has an infinite number of terms.
     
  4. Feb 26, 2013 #3
    it has been a long time since I studied Ring Theory, but here is what I remember that might be relevant:

    A unit is an element that has an inverse. So in order for ##x^2-1## to be a unit, there would have to exist an inverse of ##x^2-1## in your field. Your suggestion of ##\frac{1}{x^2-1}## is a reasonable candidate, but I do not believe it is an element of your field. This is because I always took ##F[x]## to represent the ring of finite degree polynomials over the field, F, and the Taylor expansion of ##\frac{1}{x^2-1}## is infinite.

    In my opinion, the answer needs to be "no." I think proving this hinges on the fact that we're in a field (and hence an integral domain) so there is no terms that can multiply by ##x^2## to make the leading coefficient zero.

    Good Luck!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Unit in a ring (abstract algebra)
Loading...