- #1

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## Homework Statement

Prove that ##\mathbb{Q} \times \mathbb{Q}## is not cyclic.

## Homework Equations

## The Attempt at a Solution

For contradiction suppose that ##\mathbb{Q} \times \mathbb{Q}## is cylic. Hence it is generated by some element ##(r,q)## where ##r \ne 0## and ##q \ne 0##. Then for some ##k \in \mathbb{Z}##, ##(0,q) = k \cdot (r,q) = (kr,kq)##. So ##kr = 0## and ##kq = q##. So ##k = 1##, which implies that ##r = 0##, a contradiction.

Is this proof correct? Is there a better proof, perhaps a direct one?