Algebraic Number Theory Fall 2014 These are notes for the graduate course Math 6723: Algebraic Number Theory taught by Dr. David Wright at the Oklahoma State University (Fall 2014). The notes are taken by Pan Yan (
[email protected]), who is responsible for any mistakes. If you notice any mistakes or have any comments, please let me know. Contents 1 Introduction I (08/18) 4 2 Introduction II (08/20) 5 3 Introduction III (08/22) 6 4 Introduction IV (08/25) 7 5 Group Rings, Field Algebras, Tensor Products (08/27) 9 6 More on Tensor Products, Polynomials (08/29) 11 7 Discriminant, Separable Extensions (09/03) 12 8 Trace and Norm, Commutative F -algebras (09/05) 13 9 Idempotent and Radical (09/08) 16 10 Integrality (09/10) 17 11 Noetherian Rings and Modules (09/12) 18 12 Dedekind Domains I (09/15) 19 13 Dedekind Domains II (09/17) 21 1 14 Dedekind Domains III (09/19) 23 15 Chinese Remainder Theorem for Rings(09/22) 23 16 Valuation (09/24) 24 17 Ideal Class Group in a Dedekind Domain (09/26) 25 18 Extensions of Dedekind Domain I (09/29) 26 19 Extensions of Dedekind Domain II (10/01) 28 20 Extensions of Dedekind Domain III (10/03) 29 21 Valuation Theory I (10/06) 31 22 Valuation Theory II (10/08) 32 23 Valuation Theory III (10/10) 34 24 Valuations of a Function Field (10/13) 34 25 Ostrowski's Theorem I (10/15) 36 26 Ostrowski's Theorem II (10/17) 37 27 Weak Approximation Theorem (10/20) 39 28 Completions of Valued Fields I (10/22) 41 29 Completions of Valued Fields II, Inverse Limits(10/27) 43 30 Compactness (10/29) 45 31 Hensel's Lemma (10/31)