# 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 $$mathbb{F}_{p}((t))(t^{1/p^{infty}})$$?