Prove that $Z[√5] = \{a + b√5 | a, b ∈ Z\}$ is an integral domain.
Prove that $Z[√3i] = \{a + b√3i | a, b ∈ Z\}$ is an integral domain.
I'm trying to understand how to show that these are true. By use of the definition of integral domain, I think that first I would have to show that each of these are rings under addition and multiplication. Then show multiplication is also commutative. Then show that the semigroup with multiplication has an identity element (unity). I have already done these things.
My lack of understanding is with zero divisors. $∀x,y∈D:x∘y=0_D⟹x=0_D or y=0_D$ This is the definition I have but I don't know how to apply to the two given problems above.
Any help is appreciated.
Best Answer
To check zero divisors in $\Bbb Z[\sqrt{5}]$, proceed as
$$(a+\sqrt{5}b)\cdot (c+\sqrt{5}d)=0.$$ Applying norm to both sides $$(a^2+5b^2)\cdot(c^2+5d^2)=0$$ $$\Leftrightarrow a^2+5b^2=0 \text{ or } c^2+5d^2=0$$ $$\Leftrightarrow a=b=0 \text{ or } c=d=0.$$
This shows that one of $a+\sqrt{5}b$ or $c+√5d$ is zero, which leads to $\Bbb Z[\sqrt{5}]$ is free from nontrivial zero divisors, which means $\Bbb Z[\sqrt{5}]$ is an integral domain.
Note:Similar arguments can be made in case of other rings like these.