Fields and Galois Theory - J.S. Milne   Top
Course Notes
Group Theory
Fields and Galois Theory
Algebraic Geometry
Algebraic Number Theory
Modular Functions and Modular Forms
Elliptic Curves -- see books.
Abelian Varieties
Lectures on Etale Cohomology
Class Field Theory
Complex Multiplication
Basic Theory of Affine Group Schemes
Lie Algebras, Algebraic Groups, and Lie Groups
Reductive Groups

pdf file for the current version (4.53)
pdf file formatted for ereaders (9pt; 89mm x 120mm; 5mm margins)(v4.30)

A concise treatment of Galois theory and the theory of fields, including transcendence degrees and infinite Galois extensions.


  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.


Group theory (for example, parts of my online notes), basic linear algebra, and some elementary ring theory.


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.


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. pdf file for 4.00
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. pdf file for 4.21
v4.22 (March 30, 2011). Minor changes; changed TeX style; 126 pages. pdf file for 4.22
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.