The Ramification Index-Residual Degree Identity

I have to admit that the theory of algebraic integers is so difficult and beautiful that it makes me really happy when I gain new knowledge, though it is small. I have a chance to look back and understand much more about commutative algebra, and how it is important in the theory of numbers. I have to say that it is interesting for me to see how new knowledge can be applied to give a new view of old knowledge, and I love writing. It is a time for me to digest new understanding and look back the old one.

We now come to the very important identity in algebraic number theory, the relation between ramification index and residual degree. One may care about when is a prime in $\mathbb{Z}$ is still prime in the ring of integers, and if it is not prime, how it can be factorized. It is the starting point of the topic.