Show that \((\mathbb Q,+)\) is not finitely generated as an abelian group.
Solution
Suppose \[ \mathbb Q=\langle a_1/b_1,\dots,a_r/b_r\rangle \] with \(a_i\in\mathbb Z\) and \(b_i>0\). Let \[ M=\operatorname{lcm}(b_1,\dots,b_r). \] Every integer linear combination of the proposed generators lies in \[ \frac1M\mathbb Z. \] But \[ \frac1{2M}\notin\frac1M\mathbb Z, \] since \(1/(2M)=k/M\) would imply \(1=2k\) for some \(k\in\mathbb Z\). This contradiction proves that \((\mathbb Q,+)\) is not finitely generated.