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.
This is not a complex field.
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,
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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 don’t know the definition of GCD in this context.