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.
These are the Gaussian integers .
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.
We require that,
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 .
But 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 . So I donít know the definition of GCD in this context.