Notes on Galois Theory

Notes on Galois Theory


Sudhir R. Ghorpade

Department of Mathematics
Indian Institute of Technology, Bombay
Powai, Mumbai 400076 India

Version 1.1, October 1994


These notes are aimed at providing a geodesic, yet complete, presentation starting from the definition of field extensions and concluding with the Fundamental Theorem of Galois Theory. Some additional material on separable extensions and a section on Norms and Traces is also included, and some historical comments appear as footnotes. Originally, these notes were prepared to partly supplement the lectures of the author on Galois Theory and Ramification Theory at the All India Summer School in Number Theory held at Pune in June 1991. Subsequently, the first 6 sections of the Pune Notes were separated and slightly revised to form these ``Notes on Galois Theory'', which were used for pre-conference distribution to the participants of the NBHM sponsored Instructional School on Algebraic Number Theory (University of Bombay, December 1994) at the request of the organisers. A few minor revisions have taken place in the subsequent years.


1 Preamble 2
2 Field Extensions 3
3 Splitting Fields and Normal Extensions 6
4 Separable Extensions 9
5 Galois Theory 11
6 Norms and Traces 16

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