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Math Help - Fields

  1. #1
    Newbie
    Joined
    Sep 2009
    Posts
    1

    Fields

    let F4 = (0,1,a,b) be a field containing 4 elements.
    Assume 1+ 1 = 0
    Prove that b = a^-1 = a^2 = a + 1

    What i did:

    if b = a^-1
    then b.a = (a^-1).(a)
    so ab = 1

    How can i prove the whole in equality?
    Where do i go from here?
    Any advice will be helpfull.

    Thanks
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  2. #2
    Senior Member
    Joined
    Nov 2008
    Posts
    394
    Quote Originally Posted by shing View Post
    let F4 = (0,1,a,b) be a field containing 4 elements.
    Assume 1+ 1 = 0
    Prove that b = a^-1 = a^2 = a + 1

    What i did:

    if b = a^-1
    then b.a = (a^-1).(a)
    so ab = 1

    How can i prove the whole in equality?
    Where do i go from here?
    Any advice will be helpfull.

    Thanks
    The structure of a finite field of order 4 is shown in here. It is \mathbb{Z}/2\mathbb{Z}[\alpha]/(\alpha^2 + \alpha + 1 ). Thus, a finite field of order 4 is \{0, 1, \alpha, \alpha + 1\}, where \alpha^2 + \alpha + 1 = 0 . Now you can compute the multiplicative inverse of \alpha and \alpha + 1. Note that -1=1 in \mathbb{Z}/2\mathbb{Z}.

    The structure of the above is given by the fact that
    "For any prime power q=p^n, F_q is the splitting field of the polynomial f(T) = T^q - T over F_p".
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