Thread: Gcd in Gaussian integers

1. 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.

2. 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.

3. Originally Posted by Plato I do not know what GCD in a complex field
This is not a complex field.
These are the Gaussian integers $\displaystyle \mathbb{Z}[i]$.
...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,
$\displaystyle z_1=qz_2+r$
We require that,
$\displaystyle 0\leq |r|< |z_2|$.

But an not sure what approach is taken on the division algorithm.

4. 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 $\displaystyle d(a + bi) = a^2 + b^2$.
But $\displaystyle d(11 + 4i) = d(4 - 11i)$ 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 $\displaystyle d(11 + 4i) = d(4 - 11i) = 137$. So I dont know the definition of GCD in this context.

5. 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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