MTH419: Number theory

        [Cr:4, Lc:3, Tt:1, Lb:0]

Course Outline

• Congruences, theorems of Chevalley and Warning, absolutely irreducible polynomials modulo primes.
• $p$-adic numbers, convergence in $p$-adic metric, Ostrowski's theorem.
• Valuations, local fields, structure of multiplicative group of local fields.
• Quadratic forms over $p$-adic numbers, Hilbert symbol, Hasse-Minkowski theorem.
• Characters of finite Abelian groups, Dirichlet series, Zeta function and $L$-functions, Dirichlet density theorem, Dirichlet's Theorem on primes in an arithmetic progression.

Recommended Reading

• Z. I. Borevich and I. R. Shafarevich, Number Theory, Academic Press (1966).
• Jean-Pierre Serre, A course in Arithmetic, Springer-Verklag Graduate Texts in Mathematics, Vol. 7 (1973).
• Kenkichi Iwasawa, Local Class Field Theory, Oxford University Press (1986).