How do I show that every non-zero ideal in ${\mathbb{Q}}_p$ is of the form $p^n{\mathbb{Q}}_p$ for some $n \in \mathbb{N}_0$, and investigate if ${\mathbb{Q}}_p$ is a principal ideal domain?
If it is a field it has to be an PID, too right? But I have no approach for ${\mathbb{Q}}_p$ to be from the form $p^n{\mathbb{Q}}_p$.
Try to show that it is a local ring (the only maximal ideal is $I=(p)$ since $R\setminus I$ is the set of all units).Now, every element in I is uniquely of the form p^n(a/b) where (p,a)=1 and (a,b)=1.Let I' be an ideal(not maximal).As{ a/b ,(p,a)=1} are the units in R, there is a minimal n' s.t p^n' is in I' but p^(n'-1) is not. Thus, I'=(p^n').