Motivation for considering the upper numbering of ramification groups 3 The short answer is that lower ramification groups behave well when taking subgroups, while upper ramification groups behave well when taking quotients As a result, the upper numbering can be defined for infinite extensions I think this is the key motivation for defining them
Understanding Ramification Points - Mathematics Stack Exchange I really don't understand how to calculate ramification points for a general map between Riemann Surfaces If anyone has a good explanation of this, would they be prepared to share it? Disclaimer:
How we can know the ramification ideals geometrically? How to actually compute the ramification index and inertia degree in practice is a whole other matter—my answer is just meant to give the abstract connection to geometry and some intuition about how it relates to the classical notion of ramification
what does it mean for a prime at infinity to ramify? The above definition of ramification for real places is the usual one, justified e g by the ramification index 2 which appears in a complex valuation over a real one (see Joequinn's answer) However the same phenomenon could also be interpreted as the splitting of the real place under the complex one
Understanding the Inertia Group in Ramification Theory Yes And i know about the fundamental identity But I don't know exactly how is the inertial degree related to the inertia group, even though I can imagine it being the galois group of the extension of residue fields or something like this