# Math Help - integral domain w/ complex #'s

1. ## integral domain w/ complex #'s

Let C={a+bi|a,b in Z}. Show that C is an integral domain.

Please show details of this proof. I'm very confused and this is a practice problem for my final exam.

Thank you!

2. Originally Posted by mpryal
Let C={a+bi|a,b in Z}. Show that C is an integral domain.

Please show details of this proof. I'm very confused and this is a practice problem for my final exam.

Thank you!
Hi mpryal.

It is enough to show that $C$ is a subring of the complex field $\mathbb C=\{a+bi:a,b\in\mathbb R\}$ containing 1. Any subring of a field containing the multiplicative identity is automatically an integral domain.

3. More directly.
$(a+bi)(c+di)=ac-bd + (ad+bc)i=0$

1)
$ac=bd$

2)
$ad=-bc$

multiply 1) by d on both sides

$acd=bd^2$
Apply 2
$-bc^2=bd^2$
$b(c^2+d^2)=0$

Case1) b=0 then your first one a+bi is just an integer which is definitely not a zero divisor, I think you can check that easily enough

Case2) c^2+d^2=0 but this means both c and d must be 0 because integers squared are nonnegative. But that is exactly the condition that the second gaussian integer was 0.

4. Originally Posted by mpryal
Let C={a+bi|a,b in Z}. Show that C is an integral domain.

Please show details of this proof. I'm very confused and this is a practice problem for my final exam.

Thank you!
Here is another way. You need to know that $|z_1z_2| = |z_1||z_2|$ where $z_1,z_2\in \mathbb{C}$.

If $(a+bi)(c+di) = 0 \implies |(a+bi)(c+di)| = 0 \implies |a+bi|\cdot |b+di| = 0 \implies$ $(a^2+b^2)(c^2+d^2)=0$.

Thus, $a^2+b^2=c^2+d^2 = 0 \implies a=b=c=d=0$.

EDIT: It should be "a=b=0 OR c=d=0" not "a=b=0 AND c=d=0".

5. Originally Posted by ThePerfectHacker
If $(a+bi)(c+di) = 0 \implies |(a+bi)(c+di)| = 0 \implies |a+bi|\cdot |b+di| = 0 \implies$ $(a^2+b^2)(c^2+d^2)=0$.

Thus, $a^2+b^2=c^2+d^2 = 0 \implies a=b=c=d=0$.
Hi ThePerfectHacker.

That’s not the right conclusion. What it implies is that either $a^2+b^2=0$ (so $a+ib=0)$ or $c^2+d^2=0$ (so $c+id=0).$

6. ThePerfectHacker