This is an introduction to the algebraic theory of numbers. The fundamental
techniques of the subject will be accompanied by the study of examples
of families of Diophantine equations that motivated the development of the subject.
Each of the topics listed below will occupy roughly two weeks of course time:
-
Algebraic integers, factorization, Dedekind rings, local rings
(Gauss's first proof of quadratic reciprocity) -
Units and class groups (Pell's equation, classification of binary quadratic forms)
-
Cyclotomic fields (Fermat's last theorem for regular primes, first case;
Gauss's fourth proof of quadratic reciprocity) -
Congruences and p-adic numbers (the Chevalley-Warning theorem)
-
Zeta and L-functions (Dirichlet's theorem on primes in an arithmetic progression)
-
Other topics (depending on class interest: Dirichlet's unit theorem, the prime number theorem,
cubic equations…)
Prerequisites: Basic algebra through Galois theory. Some elements of complex analysis may be admitted in section 5.
Textbook: Marc Hindry, Arithmetics (Springer, 2011 edition)
Other useful references include
Dan Flath, *Introduction to Number Theory*
Pierre Samuel, *Algebraic Theory of Numbers*
Jean-Pierre Serre, *A Course in Arithmetic*The grade will be based on homework (20%), the midterm (30%), and the take-home final (50%).