Residue field of an infinite extension of $\mathbb{Q}_p$

Question

Let $\zeta_1 \in \overline{\mathbb{Q}_p}$ such that $\zeta_1^p=p$, now let $\zeta_2 \in \overline{\mathbb{Q}_p}$ such that $\zeta_2^p=\zeta_1$ and so on with $\zeta_i^p = \zeta_{i-1}$.
I have to show that the resiude field of $K=\mathbb{Q} [ \zeta_1 , \zeta_2 , \dots ]$ is $\mathbb{F}_p$.
I noticed that $K = \bigcup_{i} \mathbb{Q}[\zeta_i]$ and I think that for every finite subfield $F$ of $K$ I can say that $F/\mathbb{Q}_p$ is totally ramified and so I can say that $K/\mathbb{Q}_p$ is totally ramified but I don't know if this is enough to show that the residue field of $K$ is $\mathbb{F}_p$, if not how can I prove that?

Answer 1

Suppose the residue field is strictly larger than $\mathbb{F}_p$. Suppose the image of $\alpha \in \mathcal{O}_K$ is not in $\mathbb{F}_p$.

Now $\alpha$ lives in some finite extension $\mathbb{Q}_p(\zeta_i)$. But you agreed that the residue field doesn't extend for any finite extensions; contradiction.