Compute explicitly the group of automorphisms $\textrm{Aut}(\mathbb{Q}^n)$ of the abelian group $\mathbb{Q}^n=\mathbb{Q}\times ...\times \mathbb{Q}$.
This task is from on an old exam. As far as I know, for $n=1$ we have $\phi(q)=q\cdot\phi(1) = q\cdot a$ for any $q\in \mathbb{Q}$ and some $a\in \mathbb{Q}$. Hence, I presume that in the general case we have $$\phi(q_1,q_2,...,q_n)=(q_1,...,q_n) \cdot \phi(1,...,1)$$ So any automorphism would be determined by how it maps $$\phi(1,...,1)=\phi(1,0,...,0)~+~...~+~\phi(0,...,0,1)$$ i.e. how it maps the basis elements. What can we conclude? Is there any meaningful group isomorphic to $\textrm{Aut}(\mathbb{Q}^n)$? How would I determine this? I'm not sure if I understand what the task is actually asking for.
Let $V$ be a $n$-dimensional vector space over $K$. Then we have $$ \operatorname{Aut}(V)\cong GL(V), $$ or in other words, $$ \operatorname{Aut}(K^n)\cong GL_n(K). $$ Here $K=\Bbb Q$.