Prove or disprove: $ \mathbb{Q} \times \mathbb{Q[i]}$ is integral domain

Question

Is $ \mathbb{Q} \times \mathbb{Q[i]}$ an integral domain ?

My attempt : I know that $ \mathbb{Q} \times \mathbb{Q}$ is not integral domain take $(0,1) \times (1,0) =( 0,0)$

But im confused in $ \mathbb{Q} \times \mathbb{Q[i]}$

@ÍgjøgnumMeg I mean, it's not very high effort, but they have shown some individual work here. More than many others. — Arthur, Dec 14 '19 at 10:14

score 4 · Accepted Answer · answered Dec 14 '19 at 10:14

4

The same example works. Here's another $(0, i) \times (1, 0) = (0, 0)$.

answered Dec 14 '19 at 10:14

badjohn

8,854

score 2 · Answer 2 · answered Dec 14 '19 at 10:15

You may use same argument to show $\mathbb{Q}\times \mathbb{Q}[i]$ is not an integral domain as $(q,0)\times (0,q')=(0,0),$ for any two non zero rational number $q,q'.$

Infact if $R$ and $R'$ are fields even, $R\times R'$ can never become integral domain.

Prove or disprove: $ \mathbb{Q} \times \mathbb{Q[i]}$ is integral domain

2 Answers2