Let $A$ be an abelian group such that $End_{Ab}(A)$ is a field of characteristc 0,prove $A \cong \Bbb{Q}$

Question

Let $A$ be an abelian group such that $End_{Ab}(A)$ is a field of characteristc 0,prove $A \cong \Bbb{Q}$

this is a exercise at p313,aluffi's algebra chapter0.And the hint is that prove A carries a structure of $\Bbb{Q}$-vector space,and look at its dimension.

Well I think I understand the fist step of the hint ,that is ,the ring of Endmorphism of A has a copy of $\Bbb{Q}$, so it carries a structure of vector space over Q.But I cant prove it has dimension 1...

Any hint will be helpful..thanks

Answer 1

Hint:If the $\mathbb{Q}$-dimension is >1, take two elements of $A$, $e_1,e_2$ which are $\mathbb{Q}$-linearly independent, consider a non trivial nilpotent linear map $B$ of $Vect_{\mathbb{Q}}(e_1,e_2)$ and show that it induces a nilpotent element of $End_{Ab}(A)$.