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Field Theory

  1. Feb 7, 2008 #1
    1. The problem statement, all variables and given/known data

    Show that F[x]/( g(x) ) is a n-dimensional vector space. where g is in F[x],
    and g has degree n.

    Its clear that F[x]/( g(x) ) is a vector space and that

    B= (1,[tex]x^{2}[/tex],.....,[tex]x^{n-1}[/tex]) spans F[x]/( g(x) ),

    but im having trouble showing that B is linearly independent

    I realize this is pretty much a HW problem and it should be in the HW section, but I
    read a post from one of the pf mentors noting that for gradlevel/seniorlevel problems
    you might have a chance at a response from the non hw sections. thanks for any suggestions.
  2. jcsd
  3. Feb 7, 2008 #2


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    Staff Emeritus
    Science Advisor
    Gold Member

    Well, what happens if they are linearly dependent, so that a nontrivial linear combination of them is equal to zero in F[x] / (g(x))?
  4. Feb 7, 2008 #3
    It's not clear that you have tied B to either F[x] or g(x). First relate B to F and g. Assume for the moment that I am not the person who doesn't have the answer.
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