I am thinking about how much information ramification properties hold about the prime numbers.
Let $\mathfrak{p}$ be a prime number in a number ring $\mathcal{O}_K$. Let $L$ be a finite extension of $K$. Factor $\mathfrak{p} = \prod_{i = 1}^g \mathfrak{q}_i^{e_i} \mathcal{O}_L$, and write $f_i = [\mathcal{O}_L / \mathfrak{q}_i : \mathcal{O}_K / \mathfrak{p}]$. The values $f_i$ are commonly known as the ramification indices. $f_i, e_i$, and $g$ depend on the number field $K$.
Can you recover $\mathfrak{p}$ from knowledge of $g$, for each number field $K$?
Can you recover $\mathfrak{p}$ from the values of $e_i$, $f_i$, and $g$ for each number field $K$?
If 2 is false, then can I have an example of two primes with the same values of $g$, $e_i$, and $f_i$ for each extension of number fields $L/K$?