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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- Jan 1st 2007, 11:43 AMJenny20Gcd 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. - Jan 1st 2007, 12:24 PMPlato
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.

- Jan 1st 2007, 01:41 PMThePerfectHacker
This is not a complex field.

These are the Gaussian integers $\displaystyle \mathbb{Z}[i]$.

Quote:

...means because the complex numbers are not usually thought to be ordered.

$\displaystyle z_1=qz_2+r$

We require that,

$\displaystyle 0\leq |r|< |z_2|$.

But an not sure what approach is taken on the division algorithm. - Jan 1st 2007, 02:51 PMPlato
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. - Jan 2nd 2007, 10:30 AMJenny20
Thank you very much .

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