Does Z[root(-3)] Qualify as a GCD Domain?

[bw0young0math]

Hello. Today I study domains but I have some problems. Please help me.

I learned Z[root(-3)] is not UFD.
Then Z[root(-3)] is a example about gcd-domain( there exist some elts a,b in domain D such that it does not exist gcd(a,b).)

Let a=2, b-1+root(-3) .
a and b are not associates and they just have common divisor 1 and -1.
So gcd(a,b) does not exist.

Hum... I don't know why associate, 1/-1 have some relations with not existing gcd. Please......

 

[Opalg]

Hello. Today I study domains but I have some problems. Please help me.

I learned Z[root(-3)] is not UFD.
Then Z[root(-3)] is a example about gcd-domain( there exist some elts a,b in domain D such that it does not exist gcd(a,b).)

Let a=2, b=1+root(-3) . I have changed a "-" to an "=" here. I hope that is what was intended.
a and b are not associates and they just have common divisor 1 and -1.
So gcd(a,b) does not exist.

Hum... I don't know why associate, 1/-1 have some relations with not existing gcd. Please......

There is something wrong here. The only common divisors of a and b are 1 and -1, and that means that a and b do have a gcd, namely 1.

If you want two elements of Z[√(-3)] without a gcd, then I suggest x = 4 and y = -2+2√(-3). Then a and b are both factors of x and of y, because
x = 2*2 = (1+√(-3))(1-√(-3)) and
y = 2(-1+√(-3)) = (1+√(-3))(1+√(-3)).
If x and y had a gcd then it would have to be a multiple of ab = 2+2√(-3). But ab does not divide x or y.