Ramification of primes on $\overline{\mathbb Q}$

Question

My concern arises about the existence of unramified primes in infinite extensions of $\mathbb Q$, particularly in $\overline{\mathbb Q}$.

Now, in general there exist be a definition of ramification for the ring $\mathbb E$ the all algebraic integers in $\overline{\mathbb Q}$?

In this MO post is defined the descomposition group of primes in infinite extensions.

Now, suppose that $p$ is a prime number and $\mathfrak p$ is a prime ideal of $\mathbb E$ such that $\mathfrak p\cap \mathbb Z=(p)$ (exist infinitely prime ideals with this feature), and define $k(\mathfrak p)=\mathbb E/\mathfrak p$, which is a algebraic clousure of $\mathbb F_p$.

Let $G(\mathfrak p)$ and $I(\mathfrak p)$ the descomposition group and the inercia group respectively of $\mathfrak p$, both respect to the absolute Galois group of $\mathbb Q$.

This gives the following sequence

$$0\rightarrow I(\mathfrak p) \stackrel{ }{\rightarrow} G(\mathfrak p) \stackrel{}{\rightarrow} Gal(k(\mathfrak p)/\mathbb F_p) \rightarrow 0$$

1. My first question is about the definining a unramified prime number.

A prime number $p$ is unramified in $\mathbb E$ if the inercia group $I(\mathfrak p)$ is trivial

Now we have to see that the definition is not empty.

2. My second question is how to prove that the definition is not empty? i.e there exist prime numbers $p$ such that $I(\mathfrak p)$ is trivial.

Thanks you all.