Topic outline

  • General

    Welcome to the Algebraic Number Theory (ANT) course!

    The aim of this course is to provide the students with the basic tools of Algebraic Number Theory and to understand how these can be applied to solve several classical problems of arithmetic nature, in special Diophantine equations. As a pretext to introduce the tools and notions, the goal of the course is to understand Kummer's proof of Fermat's Last theorem for the so-called regular primes. Prerequisites are Linear Algebra and Abstract Algebra. Galois Theory will be highly advisable but not a mandatory prerequisite. 

    Syllabus
    -Algebraic number fields and rings of integers 
    -Norms, traces and discriminants 
    -Cyclotomic and quadratic number fields 
    -Factoring elements in rings of integers 
    -Factoring ideals in the ring of integers, prime decomposition
    -Geometry of numbers: lattices and Minkowski's Theorem 
    -The ideal class group
    -The unit group and Dirichlet's Theorem 
    -Ramification and inertia 
    -Fermat's Last Theorem for regular primes 

    Bibliography
    -I. Stewart and D. Tall. Algebraic number Theory and Fermat's last theorem. Chapman and Hall 

    Assessment 
    4 mandatory sets of homework assignments, each consisting on a set of 10 exercises of theoretical and practical nature addressing different features of the course. In-class problem solving performance will also be taken into account. Grading is 1-5.