I am trying to understand why $\mathbb{Q}_3(\zeta_3)=\mathbb{Q}(\zeta_3)_{\mathfrak{P}}$ where $\mathfrak{P}$ is the unique prime above $3$ in $\mathbb{Q}(\zeta_3)$. I think I can see why this is true by looking at the degrees of these extensions over $\mathbb{Q}_3$.
However, I would like to envision the exact bijection between these fields and I find it hard to understand what elements of $\mathbb{Q}(\zeta_3)_{\mathfrak{P}}$ look like. Is it any series description for them as in the case in $\mathbb{Q}_3$?
As mentioned in the comments $\mathbb{Q}(\zeta_3)_{\mathfrak{P}}$ denotes the completion at the prime $\mathfrak{P}$.