Find gcd(137, 37+i) in Gaussian integers.
(hint: it is not 1)
How should I solve this question? Thank you very much.
This is not a complex field.
These are the Gaussian integers $\displaystyle \mathbb{Z}[i]$.
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,...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.
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.