$\begingroup$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]}$
$\endgroup$22 Answers
$\begingroup$The same example works. Here's another $(0, i) \times (1, 0) = (0, 0)$.
$\endgroup$$\begingroup$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.
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