### MA6490 Introduction to Algebraic Number Theory

#### Course Details

Commutative Algebra (General Theory of Dedekind Domains): Localization, Integral Dependence, DVR and Dedekind Domains, Modules over Dedekind Domains, Decomposition of primes in extensions. Class Groups and Dirichlet's Unit Theorem: Class groups, Norms and Traces, Discriminant, Norms of Ideals, Computing ring of integer in Quadratic Fields and Cyclotomic Fields, Quadratic Reciprocity, Minkowski's Bound, Dirichlet's Unit Theorem. Zeta and L-functions and applications: L-functions, Riemann Zeta functions, Dedekind Zeta functions, Class number formula, Frobenius Density Theorem, Dirichlet L-functions, Dirichlet's theorem on arithmetic progression. Theory of p-adic numbers: Valuations, Non-Archimedean absolute values, Completion and local fields, p-adic integers, Hensel's Lemma and applications, Ostrowski's theorem.

#### Course References:

**TextBooks : ** Gerald J. Janusz: Algebraic Number Fields, Second Edition. Graduate Studies in Mathematics, Volume 7. (American Mathematical Society, 1996)

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Reference Books: **Daniel A. Marcus, Number Fields,Springer, New York (1977). Universitext

Serge Lang, Algebraic Number Theory, Second Edition. Graduate Texts in Mathematics, Volume 110 (Springer, New York, 1994) J. Neukirch,

Algebraic Number Theory. Grundlehren Math. Wiss. (Springer, Berlin, 1999)

J.W.S. Cassels and A. Frohlich, Algebraic Number Theory, Second Edition. London Mathematical Society, 2010.