Prove that the ring of formal power series over a field is an UFD

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.

Show that each ideal $\neq 0$ has the form $\mathfrak a = t^k\mathbb R[[t]]$.
• @Georges: We don't need to use this fact. From the ideal structure one can immediately check UFD, with the only prime element $t$. – Martin Brandenburg Apr 27 '13 at 9:11
• @ramanujan_dirac Every power series $a_0+a_1t+\dots$ with $a_0\ne0$ is invertible in $\mathbb{R}[[t]]$; this should answer both your two new questions. – egreg Apr 27 '13 at 9:23