Elements of $\mathbb{Q}_p (p^{1/p^\infty})$

Question

How would one describe the field $\mathbb{Q}_p (p^{1/p^\infty})$ in terms of its elements? I know that $\mathbb{Q}_p$ is the $p$-adic completion of $\mathbb{Q}$ in the non-Archimedean norm $|\cdot|_p$; thus, we can describe the elements in terms of formal power series. But I'm not certain what adjoining $p^{1/p^\infty}$ does to this field. I saw it as an example of a "perfectoid field" in: https://arxiv.org/pdf/1303.5948v1.pdf

Answer 1

$p^{1/p^{\infty}}$ is shorthand for $p^{1/p},p^{1/p^2},p^{1/p^3},\dots$ So $\mathbb{Q}_p(p^{1/p^{\infty}}) = \bigcup_n \mathbb{Q}_p(p^{1/p^n})$. You might want to also read the AMS survey "What is a Perfectoid Space?"