TITLE: Prove that the ring of formal power series over a field is an UFD
QUESTION [6 upvotes]: The problem is from Artin. 

Prove that the ring $\mathbb{R}[[t]]$ of formal power series given by $p(t)=a_0 + a_1 t+ a_2 t^2 + \cdots$ is an UFD.

I have no idea how to do this. From the couple of things that I know about UFDs is that I could show that every irreducible element is prime, or I could show that every chain of ideals terminates. Any help will be appreciated.

REPLY [7 votes]: Show that each ideal $\neq 0$ has the form $\mathfrak a = t^k\mathbb R[[t]] $.