Why $mathbb{Z}_{p}[p^{1/p^{infty}}]/p$ $cong$ $mathbb{F}_{p}[t^{1/p^{infty}}]/t$?

I’m reading Peter Scholze’s Perfectoid Space, and I’m confused with the isomorphism $mathbb{Z}_{p}(p^{1/p^{infty}})/p$ $cong$ $mathbb{F}_{p}(t^{1/p^{infty}})/t$. What is the meaning of $mathbb{Z}_{p}(p^{1/p^{infty}})/p$ and $mathbb{F}_{p}(t^{1/p^{infty}})/t$? What’s more, how to define the completions of $mathbb{Q}_{p}(p^{1/p^{infty}})$ and $mathbb{F}_{p}((t))(t^{1/p^{infty}})$?