Fields and Galois Theory - J.S. Milne Top

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

Errata

pdf file for the current version (4.52) (11pt; a4paper; margins)

Same file with margins cropped --- may be better for viewing on gadgets.

pdf file formatted for ereaders (9pt; 89mm x 120mm; 5mm margins)(v4.30)### Contents

### Prerequisites

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

Same file with margins cropped --- may be better for viewing on gadgets.

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.

- Basic definitions and results.
- Splitting fields; multiple roots.
- The fundamental theorem of Galois theory.
- Computing Galois groups.
- Applications of Galois theory.
- Algebraic closures
- Infinite Galois theory
- The Galois theory of étale algebras
- Transcendental Extensions.

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.