Instructor : Anupam Kumar Singh, IISER Pune
Schedule : January - April 2011
Prerequisite : Rings, Modules, Field Extensions, Galois Theory
Objective : The course is meant for 4th year students with good background in Algebra. This is a basic course in Algebraic Number Theory. The subject was developed to solve one of the most famous problems in Mathematics : Fermat's Last Theorem. In this course we will see how one can solve the problem partially using this theory and completely if you make a mistake.
Content : Number Fields, Ring of Integers, Dedekind Domains, Solving Polynomials, Integral Bases, Ideal factorization, Ideal Class Group, Finiteness of Class group, Dirichlet's Unit Theorem, Solution of FLT in a special case.
Evaluation : 3 tests (20% each) + a project (30%) + teacher's assessment (10%)
References :
1) Number Fields: Daniel A. Marcus, Universitext, Springer;
2) Problems in Algebraic Number Theory: M. RamMurty and Jody Esmonde, Graduate Texts in Mathematics, Springer;
3) Algebraic Number Theory: Serge Lang, Graduate Texts in Mathematics, Springer.
4) A brief Guide to Algebraic Number theory : Swinnerton-Dyer, LMSST 50
5) Algebraic Number Theory : Stewart and Tall
6) Number Theory : Borevich & Shaparevich
7) Local Fields : Cassels
8) A Course in Arithmetic : Serre
9) Local fields : Serre
10) Algebraic Number theory : J S milne (available online)
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Test I (11/02/11) Exercises