Lectures on Field Theory and Ramification Theory

Lectures on Field Theory and Ramification Theory


Sudhir R. Ghorpade

Department of Mathematics
Indian Institute of Technology, Bombay
Powai, Mumbai 400076 India
E-mail: srg@math.iitb.ac.in

Version 1.1, June 1995


These are the notes of a series of five lectures given at the NBHM sponsored Instructional School on Algebraic Number Theory held at University of Bombay during December 27, 1994 - January 14, 1995. These lectures were aimed at covering the essentials of Field Theory and Ramification Theory as may be needed for local and global class field theory. Most of the material in these notes, and other related material, is now available in the Kiel notes. However, the two sections on cyclic extensions and abelian extensions (Artin-Schreier, Hilbert Theorem 90 and Kummer theory) are not included in the Kiel notes.


1 Field Extensions 2
1.1 Basic Facts 2
1.2 Basic Examples 5
1.3 Cyclic Extensions 8
1.4 Abelian Extensions 11
1.5 Discriminant 13
2 Ramification Theory 20
2.1 Extensions of Primes 21
2.2 Kummer's Theorem 24
2.3 Dedekind's Discriminant Theorem 25
2.4 Ramification in Galois Extensions 37
2.5 Decomposition and Inertia Groups 29
2.6 Quadratic and Cyclotomic Extensions 31
Bibliography 35

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