# Is it true that $\mathbb{Z}[\sqrt{-3}] + \mathbb{Z}[\sqrt{-5}]$ not a ring?

Question: Is it true that $$\mathbb{Z}[\sqrt{-3}] + \mathbb{Z}[\sqrt{-5}]$$ not a ring?

I think it is not because it is not closed under multiplication. More precisely, if $$a_1+b_1\sqrt{-3}, a_2+b_2\sqrt{-3}\in \mathbb{Z}[\sqrt{-3}]$$ and $$c_1+d_1\sqrt{-5},c_2+d_2\sqrt{-5}\in \mathbb{Z}[\sqrt{-5}],$$ then $$(a_1+b_1\sqrt{-3} + c_1+d_1\sqrt{-5}) \cdot (a_2+b_2\sqrt{-3} + c_2+d_2\sqrt{-5})$$ is not an element of $$\mathbb{Z}[\sqrt{-3}] + \mathbb{Z}[\sqrt{-5}].$$

• That would be correct. – Parcly Taxel Nov 17 '18 at 4:50
• That can be an element of $\Bbb Z[\sqrt{-3}]+\Bbb Z[\sqrt{-5}]$ for some values of the variables. To show it's not closed under multiplication, all one has to do is give an example. The easy example is $\sqrt{-3}\times\sqrt{-5}$. – Lord Shark the Unknown Nov 17 '18 at 5:28

For instance, $$\sqrt{-3}$$ and $$\sqrt{-5}$$ are both in your set, but their product (whether that's $$-\sqrt{15}$$ or $$\sqrt{15}$$) isn't.
However, $$\Bbb Z[\sqrt{-3}]+\Bbb Z[\sqrt{-5}]+\Bbb Z[\sqrt{15}]$$ is a ring.