Is $\mathbb{Q}\times \mathbb{Q}$ a field?

Question

Is $\mathbb{Q} \times \mathbb{Q}$ a field ?

We know that $\mathbb Q$ is a field. But, if we think of $\mathbb Q\times \mathbb Q$, what would be the answer?

Hint: what's the inverse of $(0,a) \in \mathbb{Q}\times \mathbb{Q}$? — Alex Wertheim, Dec 04 '14 at 16:11

score 4 · Accepted Answer · answered Dec 04 '14 at 16:27

4

For every $0 \neq a,b\in K$ for a field $K$ we have that $(a,0) \cdot (0,b)=(0,0)$ , so $K \times K$ is not an integral domain, hence not a field.

answered Dec 04 '14 at 16:27

Marm

1 Answers1