If $F$ is a field, I know $F[x]$ is ED(hence PID and hence UFD). Then what happens to $F[[x]]$?
I Know every ideal of $F[[x]]$ is principal so $F[[x]]$ is at least PID. (I Know $F[[x]]$ is UFD) And $\langle X \rangle \in F[[x]]$ is maximal ideal, i.e., $F[[x]]$ is a local ring. [A commutative ring with a unique maximal ideal is called a local ring]
I want to know minimum constraints like ED, PID, UFD for $F[[x]]$. It seems $F[[x]]$ cannot be field, am I right?