Algebraic Number 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

Current version (3.07)
pdf file for printing (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)(3.03)

This is a fairly standard graduate course on algebraic number theory.


  1. Preliminaries From Commutative Algebra
  2. Rings of Integers
  3. Dedekind Domains; Factorization
  4. The Finiteness of the Class Number
  5. The Unit Theorem
  6. Cyclotomic Extensions; Fermat's Last Theorem
  7. Valuations; Local Fields
  8. Global Fields


The algebra usually covered in a first-year graduate course, including Galois theory, group theory, and multilinear algebra. An undergraduate number theory course will also be helpful.


v2.01; August 14, 1996; first version on the web; 144p.
v2.10; August 31, 1998; fixed many minor errors; added exercises and index; 140p.
v3.00; February 11, 2008; corrected; revisions and additions; 163 pages.
v3.01; September 28, 2008; fixed a problem with the hyperlinks; 163 pages.
v3.02; April 30, 2009; fixed many minor errors; changed chapter and pages styles; 164 pages.
v3.03; May 29, 2011; minor fixes; 167 pages.
v3.04; April 16, 2012; minor fixes; 160 pages.
v3.05; March, 21, 2013; minor fixes; 161 pages.
v3.06; May, 28, 2014; minor fixes; added Hermite's theorem; 164 pages.
v3.07; March 18, 2017; minor fixes; 165 pages.

pdf (old version 2.10)
pdf (old version 3.00)
pdf (old version 3.01)