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Math Help - Ring

  1. #1
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    Ring

    Let A be a ring with 0 \not = 1 and K be the set
    K ={ x\in A | x^2 - x + 1 = 0}.
    1) If x\in K then x is invertible and x^{ - 1} = x^5
    2) If K = A \ {0,1} then A is isomorphic with Z_{3} or 1 + 1 = 0 .
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  2. #2
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    Just to start you off on this, if x^2-x+1=0 then x^2=x-1, x^3=x(x-1) = x^2-x = (x-1)-x = -1.
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  3. #3
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    Quote Originally Posted by petter View Post
    Let A be a ring with 0 \not = 1 and K be the set
    K ={ x\in A | x^2 - x + 1 = 0}.

    2) If K = A \ {0,1} then A is isomorphic with Z_{3} or 1 + 1 = 0 .
    suppose 1+1 \neq 0. so -1 \in A - \{0,1\}=K. thus: 1+1+1=(-1)^2 + 1 + 1 =0. hence 1+1=-1. in order to prove that A \simeq \mathbb{Z}_3, we only need to show that K=\{-1 \}.

    so suppose -1 \neq x \in K. so x^2-x+1=0. call this (1). now since x+1 \in A - \{0,1 \}=K, we also have: (x+1)^2 - (x+1) + 1 = 0, which gives us: x^2 + x + 1 = 0. \ \ (2)

    (1) and (2) give us: -x=(1+1)x = 0, i.e. x=0. contradiction! Q.E.D.
    Last edited by NonCommAlg; January 5th 2009 at 05:31 AM.
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