$p$-adic completion of $\mathbb Z((t))$

Question

Consider the ring of formal Laurent power series $\mathbb Z((t))$ and let $p$ be a prime. Let $(p)$ be the ideal in $\mathbb Z((t))$ generated by $p$:

1. What is an explicit expression of the $(p)$-adic completion of $\mathbb Z((t))$?
2. Let $A$ be the completed ring obtained in 1. What is its fraction field?

Answer 1

$Frac(\Bbb{Z}_p[[t]])$ is a subfield of the field $$F=\{t^{-N} p^{-m} \sum_{n\ge 0} a_n p^{-nk} t^n \in \Bbb{Q}_p((t)), a_n \in \Bbb{Z}_p,N,m,k \in \Bbb{Z}\}$$

I'd say $Frac(\Bbb{Z}_p[[t]]) $ is still smaller than $F $ and that there is no easy characterization of the formal series in $ \Bbb{Q}_p((t))$ that are quotients of two elements of $Frac(\Bbb{Z}_p[[t]])$