How to understand $\mathbb{Q}_{p}(p^{1/p^{\infty}})$?

Question

It is known that $\mathbb{Q}_{p}(p^{1/p^{\infty}})$ is defined to be $\bigcup_{n>0} \mathbb{Q}_{p}(p^{1/p^{n}})$, which means adjoining all $p$-power roots of $p$ to the mixed characteristic field $\mathbb{Q}_{p}$. However, I have problem understanding the symbol $\mathbb{Q}_{p}(p^{1/p^{n}})$. How can this relate to the $p$-power roots of $p$? Why in the symbol, the power of $p$ is $1/p^{n}$? I think that $\mathbb{Q}_{p}(p^{1/p^{n}})$ is a cyclotomic extension of $\mathbb{Q}_{p}$, where $p^{1/p^{n}}$ is the primitive $n$th root of unity. But it seems that this does not make sense. And I saw in other answers that $\mathbb{Q}_{p}(p^{1/p^{n}})$ is a ramified extension. Could anyone tell me where can I learn about $\mathbb{Q}_{p}(p^{1/p^{n}})$?