Following on from my question of the proof of the ring of a FPS is an integral domain:
Proving the ring of formal power series over a finite field is integral domain.
I would like to extend to that to finding a proof of showing that the ring of formal power series over a finite field of order prime is not a field.
Just confused really, with proving it was an integral domain, it seemed to be fine.
Thanks for any help
-nomad609
You could find a noninvertible element, as the comments suggest, or you could (equivalently) find a nonzero ideal that isn't the whole ring itself. Consider the homomorphism \begin{align*} \phi : k[\![T]\!]&\to k\\ \sum_{i = 0}^\infty a_i T^i&\mapsto a_0. \end{align*} The homomorphism above is clearly surjective, and it's not hard to verify that the kernel is $(T)$. In a field, the only ideals are $(0)$ and $(1)$. But $(0)\subsetneq (T)\subsetneq (1)$ ($T\not\in (0)$, and $1\not\in (T)$, because $\phi(1) = 1$), so $k[\![T]\!]$ cannot be a field.