1. ## linearly independent of additive group

Find the linearly independent subsets of the rational additive group over $\mathbb{Z}$

Thanks!

2. Originally Posted by roporte

Find the linearly independent subsets of the rational additive group over $\mathbb{Z}$

Thanks!
$A \subset \mathbb{Q}$ is linearly independent over integers if and only if $A=\{r\},$ for some $0 \neq r \in \mathbb{Q}.$ such sets are obviously linearly independent. now suppose $B \subset \mathbb{Q}$ is linearly independent over integers and

$|B| \geq 2.$ obviously $0 \notin B.$ choose any two elements of $B$, say $r_1=\frac{a}{b}, \ r_2=\frac{c}{d}.$ then $bcr_1 - adr_2 =0,$ which proves that $\{r_1,r_2 \}$ and therefore $B$ is not linearly independent over integers. Q.E.D.