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Field maths

  1. Feb 19, 2008 #1
    1. The problem statement, all variables and given/known data
    Let F be a field. For any a,b [tex]\in[/tex] F, b[tex]\neq[/tex]0, we write a/b for ab^-1. Prove the following statements for any a, a' [tex]\in[/tex]F and b, b' [tex]\in[/tex] F\{0}:

    i.) a/b = a'/b' if and only if ab' =a'b
    ii.) a/b +a'/b' = (ab'+a'b)/bb'
    iii.) (a/b)(a'/b') = aa'/bb'
    iv.) (a/b)/a'/b') = ab'/a'b (if in addition a'[tex]\neq[/tex]0)


    2. Relevant equations



    3. The attempt at a solution

    I'm struggling to understand how i am to prove these statements. What am i to take the dashes to mean, because they are often used to show inverses? So for the first one:

    a/b=ab^-1 which = a^-1b = a'/b'?
     
  2. jcsd
  3. Feb 19, 2008 #2

    EnumaElish

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    My guess is a dash means "alternative value." For example, a = 1, and a' = 2.
     
  4. Feb 19, 2008 #3
    I also thought that, i will go with that and see what i come up with, thanks
     
  5. Feb 19, 2008 #4
    Does ab^-1 mean a.b^-1 or (a.b)^-1? I think it might be the former.

    If it is, i get:

    i) a.b^-1 = a'.b'^-1 when written out fully. So if ab' = a'b, then rearranged gives a= a'b/b' and a' = ab'/b. So inserting them into a.b^-1 = a'.b'^-1 we get:

    a'b.b^-1/b' = ab.b^-1/b

    and then we get indentity elements leaving a'/b' = a/b

    Is this right?
     
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