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Rings and Fields - Write down the nine elements of F9

  1. Sep 23, 2010 #1
    Rings and Fields - Write down the nine elements of F9

    1. The problem statement, all variables and given/known data

    In F9 = Z/3Z, there is no solution of the equation x^2 = −1, just as in R. So “invent”
    a solution, call it 'i'. Then 'i' is a new “number” which satisfies i^2 = −1. Consider
    the set F9 consisting of all numbers a+bi, with a,b in F9. Add and multiply these
    numbers as though they were polynomials in 'i', except whenever you get i^2 replace
    it by −1.
    (i) Write down the nine elements of F9 .
    (ii) Show that every nonzero element of F9 has an inverse, so that F9 is a
    field.


    3. The attempt at a solution

    I know im supposed to show you that ive tried the question if i want an answer. Believe me, i have tried it. Im just really confused by the wording of the question and am not really sure what they are looking for in part a. Once i get part a, im pretty sure id be able to get part b on my own.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 23, 2010 #2
    Re: Rings and Fields - Write down the nine elements of F9

    You are told that F9 = {a+bi | a,b in F9}. The first part is asking you to try out all the elements. So for example, 1+2i is an element.
    The second part is asking you to show that for all a+bi in F9, there is some c+di such that (a+bi)(c+di) = 1. Try writing the inverse out using a and b in some way, then multiply it out to check if you are correct and get 1.
     
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