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Math Help - Gcd in Gaussian integers

  1. #1
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    Post Gcd in Gaussian integers

    Find gcd(137, 37+i) in Gaussian integers.
    (hint: it is not 1)

    How should I solve this question? Thank you very much.
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  2. #2
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    I do not know what GCD in a complex field means because the complex numbers are not usually thought to be ordered. However, it is easy to see that both 11+4i and 4-11i divide both 137 and 37+i giving a Gaussian integer.
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  3. #3
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    Quote Originally Posted by Plato View Post
    I do not know what GCD in a complex field
    This is not a complex field.
    These are the Gaussian integers \mathbb{Z}[i].
    ...means because the complex numbers are not usually thought to be ordered.
    That is true. I never studied this section of field theory in great detail but I happen to know that the Gaussian Integers and the Polynomials form an Euclidean domain. Hence, there exists a gcd. I think (but I might be wrong) when we express,
    z_1=qz_2+r
    We require that,
    0\leq |r|< |z_2|.

    But an not sure what approach is taken on the division algorithm.
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  4. #4
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    Thank you for the input. I am aware that Z[i] or the Gaussian Integers are an Euclidian Domain in which the measure function is d(a + bi) = a^2  + b^2.
    But d(11 + 4i) = d(4 - 11i) both divisors of 137 and 37+i. These are the only two non-units that divide both. We see that in order for w to divide the real integer n in Z[i], d(w) divides n. Because 137 is prime we have d(11 + 4i) = d(4 - 11i)  = 137. So I donít know the definition of GCD in this context.
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  5. #5
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    Thank you very much .

    from 137 =4*(37+i) + (-11-4i). I got the gcd is 11+4i. I hope this is correct answer.
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