Is $\mathbb{C}[[t]]$ isomorphic to the localization of $\mathbb{C}[t]$ at the maximal ideal $(t)$?

Question

It seems to me that the power series ring $\mathbb{C}[[t]]$ is isomorphic to the localization of $\mathbb{C}[t]$ at the maximal ideal $(t)$, but I am not sure.

Answer 1

This is not true. The ring $\mathbb{C}[t]_{(t)}$ is naturally a subring of $\mathbb{C}[[t]]$, yet this inclusion is proper. For example one can solve $\sqrt{1-t}$ over $\mathbb{C}[[t]]$ and there is no solution over $\mathbb{C}[t]_{(t)}$.