I have published this in order to make it permanently available and to allow individuals and libraries to purchase a hardback copy. Cite it as this. Apart from the pagination, the course notes file v5.10 is identical to the 2022 version of the book, which is available everywhere. This file v5.10 is frozen. For corrections and possible updates, see the book.

Current version (5.10, 2022). pdf file
Version 5.00, 2021. pdf file
Version 5.00, 2021. Source files
Version 4.30 pdf file formatted for ereaders (9pt; 89mm x 120mm; 5mm margins)

I have made Version 5.0, including the source files, available under a Creative Commons licence CC BY-NC-SA 4.0. This means that the work is not only freely available, but also freely editable. Roughly speaking, the licence allows you to edit and redistribute this text in any way you like, as long as you include an accurate statement about authorship and copyright, do not use it for commercial purposes, and distribute it under this same licence.

Blurb

These notes give a concise exposition of the theory of fields, including the Galois theory of field extensions, the Galois theory of étale algebras, and the theory of transcendental extensions. The first five chapters treat the material covered in most courses in Galois theory while the final four are more advanced.

The first two chapters are concerned with preliminaries on polynomials and field extensions, and Chapter 3 proves the fundamental theorems in the Galois theory of fields. Chapter 4 explains, with copious examples, how to compute Galois groups, and Chapter 5 describes the many applications of Galois theory.

In Chapter 6, a weak form of the Axiom of Choice is used to show that all fields admit algebraic closures, and that any two are isomorphic. The last three chapters extend Galois theory to infinite field extensions, to \'etale algebras over fields, and to nonalgebraic extensions.

The approach to Galois theory in Chapter 3 is that of Emil Artin, and in Chapter 8 it is that of Alexander Grothendieck.

The only prerequisites are an undergraduate course in abstract algebra and some group theory, for example, the first six chapters of GT. There are 96 exercises, most with solutions.

Contents

1. Basic definitions and results.
2. Splitting fields; multiple roots.
3. The fundamental theorem of Galois theory.
4. Computing Galois groups.
5. Applications of Galois theory.
6. Algebraic closures
7. Infinite Galois theory
8. The Galois theory of étale algebras
9. Transcendental Extensions.

Translations

Vietnamese translation of v4.53 Translated by Dr. Le Minh Ha, Faculty of Mathematics-Mechanics-Informatics, VNU University of Science, ThanhXuan, Hanoi, Vietnam

Beautiful logo for a course on Fields and Galois Theory at Tribhuvan University, Kathmandu, Nepal.

History

v2.01 (August 21, 1996). First version on the web.
v2.02 (May 27, 1998). Minor corrections; 57 pages.
v3.0 (April 3, 2002). Revised notes; minor additions to text; added 82 exercises with solutions, an examination, and an index; 100 pages.
v3.01 (August 31, 2003). Minor corrections; numbering unchanged; 99 pages.
v4.00 (February 19, 2005). Minor corrections and improvements; added proofs to the section on infinite Galois theory; added material to the section on transcendental extensions; 107 pages.
v4.10 (January 22, 2008). Minor corrections and improvements; added proofs for Kummer theory; 111 pages.
v4.20 (February 11, 2008). Replaced Maple with PARI; 111 pages.
v4.21 (September 28, 2008). Minor corrections; fixed problem with hyperlinks; 111 pages.
v4.22 (March 30, 2011). Minor changes; changed TeX style; 126 pages.
v4.30 (April 15, 2012). Minor corrections; added sections on etale algebras; 124 pages.
v4.40 (March 20, 2013). Minor corrections and additions; 130 pages.
v4.50 (March 18, 2014). Added chapter explaining Grothendieck's approach to Galois theory (Chapter 8) and made many minor improvements; numbering has changed; 138 pages.
v4.51 (August 31, 2015). Minor corrections; 138 pages.
v4.52 (March 17, 2017). Minor corrections; 138 pages.
v4.53 (May 27, 2017). Minor fixes and improvements; 138 pages.
v4.60 (September 2018). Minor fixes and additions; numbering unchanged; 138 pages.
v4.61 (April 2020). Minor fixes and additions; Chapter 8 revised; numbering little changed; 138 pages.
v5.00 (June 2021). Made work, including the source files, available under a Creative Commons licence CC BY-NC-SA 4.0.
v5.10 (September 2022). Modest revision, especially of Chapters 2,6; some numbering changed; 144 pages.