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Homework Help: 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


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    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!
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