Reference Request: Subgroup of free abelian group is free abelian

Question

I have the following reference Question, meaning that I search for a reference for the following statement: Let $F$ be a free abelian group of finite rank and $U$ be a subgroup. Then there is a basis $\{b_1, \dots, b_n\}$ of $F$ and numbers $k_1, \dots, k_l \geq 1$ $(l \leq n)$ such that $\{k_1b_1, \dots, k_lb_l\}$ is a basis of $U$.

I found a proof in Langs Algebra book that subgroups of free abelian groups are free abelian, but not this specific computation of the basis.

Thanks for any help.

Answer 1

This is known as Smith normal form, and you can find lots of references on Google books. In particular, this textbook has exhaustive coverage of Smith normal form and all of its consequences.