Ramification Theory of Local Fields
This course introduces local fields and the ramification of their finite Galois extensions. Starting from discrete valuations and completions, it develops unramified and tame extensions, inertia and wild ramification, the different and discriminant, and the lower and upper ramification filtrations.
Cyclotomic and Artin–Schreier extensions provide explicit examples connecting field equations, Galois actions, and arithmetic invariants. The course combines core proofs with worked examples and guided problem-solving.
Cyclotomic and Artin–Schreier extensions provide explicit examples connecting field equations, Galois actions, and arithmetic invariants. The course combines core proofs with worked examples and guided problem-solving.
Lecturer
Date
19th October, 2026 ~ 6th January, 2027
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Wednesday | 13:30 - 15:05 | A14-202 | ZOOM 03 | 242 742 6089 | BIMSA |
Prerequisite
Abstract algebra and Galois theory, including rings, ideals, fields, polynomials, and finite Galois extensions. No previous knowledge of local fields or algebraic number theory is required. Valuations, Cauchy sequences, and completeness will be introduced, and the necessary facts about finite fields will be reviewed.
Syllabus
Discrete valuations and local rings
Valuations, valuation rings, units, ideals, uniformizers, and residue fields. Connections between valuations and divisibility.
Completions and Hensel’s lemma
Construction and basic properties of p-adic fields and Laurent series fields. Nonarchimedean convergence and simple-root Hensel lifting.
Finite extensions of local fields
Extension and normalization of valuations. Ramification indices, residue degrees, integral bases, and the degree formula.
Unramified and Eisenstein extensions
Construction of unramified extensions by residue-field lifting. Frobenius automorphisms, Eisenstein polynomials, and the maximal unramified intermediate field.
Tame ramification
Structure of totally tamely ramified extensions. Radical extensions of degree prime to the residue characteristic and the first cyclotomic example.
Inertia and lower ramification groups
The residue-field exact sequence, the lower ramification filtration, the uniformizer test, and compatibility with subgroups.
Wild inertia and explicit filtrations
Graded ramification quotients and the structure of wild inertia as a p-group. Computation of a wild cyclotomic ramification filtration.
The different and discriminant
Trace duality, the codifferent, and the derivative formula for extensions admitting an integral power basis. The relation between different and discriminant exponents.
Hilbert’s formula and towers
Hilbert’s different formula and transitivity of the different. Cyclotomic calculations using both ramification groups and polynomial derivatives.
Upper numbering and Herbrand’s theorem
Herbrand functions and their inverses. Upper ramification groups, compatibility with quotients, and applications to explicit towers.
Artin–Schreier extensions
Reduction of equations in characteristic p. Explicit uniformizers, total ramification, and computation of a single wild break and the different.
Synthesis and applications
Complete ramification calculations and comparison of cyclotomic and Artin–Schreier examples. Review of the main results and methods.
Valuations, valuation rings, units, ideals, uniformizers, and residue fields. Connections between valuations and divisibility.
Completions and Hensel’s lemma
Construction and basic properties of p-adic fields and Laurent series fields. Nonarchimedean convergence and simple-root Hensel lifting.
Finite extensions of local fields
Extension and normalization of valuations. Ramification indices, residue degrees, integral bases, and the degree formula.
Unramified and Eisenstein extensions
Construction of unramified extensions by residue-field lifting. Frobenius automorphisms, Eisenstein polynomials, and the maximal unramified intermediate field.
Tame ramification
Structure of totally tamely ramified extensions. Radical extensions of degree prime to the residue characteristic and the first cyclotomic example.
Inertia and lower ramification groups
The residue-field exact sequence, the lower ramification filtration, the uniformizer test, and compatibility with subgroups.
Wild inertia and explicit filtrations
Graded ramification quotients and the structure of wild inertia as a p-group. Computation of a wild cyclotomic ramification filtration.
The different and discriminant
Trace duality, the codifferent, and the derivative formula for extensions admitting an integral power basis. The relation between different and discriminant exponents.
Hilbert’s formula and towers
Hilbert’s different formula and transitivity of the different. Cyclotomic calculations using both ramification groups and polynomial derivatives.
Upper numbering and Herbrand’s theorem
Herbrand functions and their inverses. Upper ramification groups, compatibility with quotients, and applications to explicit towers.
Artin–Schreier extensions
Reduction of equations in characteristic p. Explicit uniformizers, total ramification, and computation of a single wild break and the different.
Synthesis and applications
Complete ramification calculations and comparison of cyclotomic and Artin–Schreier examples. Review of the main results and methods.
Audience
Graduate
, Postdoc
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
I have been an assistant professor at BIMSA since January 2024. My research primarily focuses on p-adic differential equations. I defended my Ph.D. thesis in 2018 at Montpellier University. Before joining BIMSA, I was an assistant professor at Algiers University. Subsequently, I was a postdoc at YMSC, Tsinghua University, from April 2021 to December 2023.