Automorphism groups of $\mathbb{Z}^n $ and $\mathbb{Q}^n $ .

Question

I was thinking about structures of $Aut( \mathbb{Z}^n)$ and $Aut( \mathbb{Q}^n)$ . I know the group $Aut(\mathbb{Z})$ and $Aut(\mathbb{Q})$ but i dont know where i need to start for a general result. It will be good to look at an article about it but i couldn't find one. Any article suggestion or hint for starting would be great.

Answer 1

$\mathbb{Z}^n$ is a free abelian group and so $Aut( \mathbb{Z}^n) \cong GL(n,\mathbb{Z})$.

$\mathbb{Q}^n$ is a vector space over $\mathbb{Q}$ and every additive automorphism of $\mathbb{Q}^n$ is actually a vector space automorphism. Therefore, $Aut(\mathbb{Q}^n) \cong GL(n,\mathbb{Q})$.