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Algebraic structure

  1. Nov 20, 2005 #1
    For Zn = { 0, 1 ,...,n-1}, the algebraic structure (Zn, +, . ) is a "ring", i.e., it has nearly all of the usual properties of addition and multiplication that we use unconsciously most of the time(where the opertaions are defined by performing them in Z and then recording the remainder on division by n). In Z, of course, the only invertible elements with respect to multiplication (a for which there is some b such that ab = 1), are +-1. PRove that the invertible elements with respect to multiplication in Zn are exactly those elements a such that a and n are relatively priime; that is , gcd{a,n}=1

    can some one give me a hint on wat to do in this problem? i woud really apriciate it!!
  2. jcsd
  3. Nov 20, 2005 #2


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    What do you know about gcd? Are you familiar with Euclid's algorithm? If not, look it up on Wikipedia.
  4. Nov 20, 2005 #3

    yes i know wat it is and i also know how to solve it.. but i dont see how am i suppost to use the euc alg to solve this problem.
    any hint?
    thank u
  5. Nov 20, 2005 #4


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    Do you know the algebraic characterization of the GCD? IMHO, it is much more useful than the one involving divisibility.
  6. Nov 20, 2005 #5

    matt grime

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    the euclidean algorithm states that if a and n are coprime there are integers x and y such that "SOMETHING THAT GIVES AWAY THE ANSWER"

    if you do know the Euclidean algorithm then the answer is obvious, surely?
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