Background
Let $K$ be a (unital, associative) commutative ring and consider the dual $K$-module $K[x]^*$ of the polynomial algebra. Then there is a $K$-algebra structure on $K[x]^*$ satisfying $$x_n \cdot x_m = \binom{n+m}{m} x_{n+m}$$ where $x_n$ denotes the dual element to $x^n$. This is the dual algebra associated to the divided power coalgebra structure on $K[x]$ (the one that defines a bialgebra structure along with the usual algebra structure on $K[x]$).
There is always a $K$-algebra map $$\Phi : K[[t]] \rightarrow K[x]^*$$ that takes $t^n$ to $n! x_n$. I have shown that $\Phi$ is an isomorphism if and only if $K$ is a $\mathbb{Q}$-algebra.
Question
What are the precise conditions on $K$ such that there exists a $K$-algebra isomorphism $K[x]^* \cong K[[t]]$?
I would really appreciate a "(co)homological" argument as well, say by considering $\mathbb{CP}^\infty$.