MTH421 Number Theory

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