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Math Help - Finding group of units

  1. #1
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    Question Finding group of units

    Q: Determine U(Z[i]) ?

    My book solves this by taking some unit a+bi of Z[i], and then multiplying it with its multiplicative inverse say, c+di.
    => (a+bi)(c+di) = 1
    Taking modulus on both sides
    => (a2+b2)(c2+d2) = 1
    => a2+b2= 1, I don't really understand this step. Can someone please tell me how my book arrived to this conclusion ?

    Please Help.
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  2. #2
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    Re: Finding group of units

    Hi,
    You're overlooking an "obvious" fact: if m and n are non-negative integers with product 1, both m and n must be 1.
    Proof.
    Neither m nor n can be 0. If m is not 1, m is greater than or equal to 2 and the product mn would also be greater than or equal to 2, a contradiction. Same for n.
    Thanks from mrmaaza123
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