1. Limited time only! Sign up for a free 30min personal 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!

Proof of associates in Zp[x]

  1. Mar 25, 2009 #1
    Show that a nonzero polynomial in Zp[x] has p-1 associates.

    I don't have a proof as much as a fairly weak (in my opinion) arguement.

    Suppose you had a series of functions with coefficients p. If the coeffients are p in Zp[x], all those functions go to zero. In this case you just have a bunch of zeroes, which can't happen since there is no zero associate in Zp[x]. I think I may have butchered this proof as we just went over associates and I'm not sure how to do this.
     
  2. jcsd
  3. Mar 25, 2009 #2

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    I'm not familiar with the term; what's an "associate"?
     
  4. Mar 25, 2009 #3
    An associate is a nonzero constant that a polynomial is multiplied by

    e.q 2x+2 is 2(x+1)...2 is the associate
     
  5. Mar 25, 2009 #4

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Er... so "an associate of a polynomial" is simply "the leading coefficient of that polynomial"? But any polynomial would have only one associate then.
     
  6. Mar 25, 2009 #5
    An associate is a nonzero constant multiple of a polynomial.
     
  7. Mar 25, 2009 #6

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Ah, okay. Well... since each associate is determined by a "nonzero constant", we can simply count those, can't we? Well, we do have to make sure that the same associate isn't counted multiple times by this method. (Or make a correction if that does happen)
     
  8. Mar 25, 2009 #7
    So here is what I got:

    Suppose yyou have a polynomial x^n....+x+c such that p is an associate:

    p*(x^n+....+x+c) => px^n+...+p*x+p*c => 0+....0+0

    This contradicts the statement.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Proof of associates in Zp[x]
  1. Proof of f(x) = x (Replies: 51)

  2. Sup(X)=-inf(-X) proof. (Replies: 0)

Loading...