Algebraic Number Theory

Syllabus Details

• Unit I: (Introduction) Motivation, Gaussian integers $\mathbb{Z}[i]$ and the ring $\mathbb{Z}[\sqrt{-5}]$, Classification of Gaussian Primes, Ideals and Lattice structures in these rings.
• Unit II: (The Ring of Algebraic Integers $\mathcal{O}_{K}$): Integral Elements and equivalent conditions for being integral, integral ring extensions, Integral closure, The ring of algebraic integers of a number field $\mathcal{O}_{K}$. Integral basis Theorem: existence and construction, Trace and Norm maps: $\mathrm{Tr}_{K/\mathbb{Q}}$ and $\mathrm{N}_{K/\mathbb{Q}}$, Discriminants, Group of Units in Imaginary Quadratic fields, Cyclotomic Integers.
• Unit III: (Dedekind Domains) Structure of the ring of integers $\mathcal{O}_{K}$ Noetherian, integrally closed and every prime ideal is maximal, Fractional ideals, Dedekind domains and equivalence properties, Uniqueness of Factorization of ideals into product of prime ideals, Ramification of Primes (in Quadratic Number Fields).
• Unit IV: (The Ideal Class Group) Definition of Ideal Class Group, Finiteness of Class Group: Minkowski’s Theorem (on lattice points) and applications.
• Unit V: (Units in the Ring of Integers) Group of Units in $\mathcal{O}_{K}$, The Dirichlet’s unit theorem (with proof).

Course Learning Objective

1. Understand integral extensions of rings with examples.
2. Study the Ring of Algebraic Integers in Number Field.
3. Understand ideal factorizations using Dedekind Domain Structure and some computations.
4. Find integral basis.
5. Understand finiteness of Ideal Class Group.
6. Understand structure of units in $\mathcal{O}_{K}$.

Recommended Books

1. Narasimhan, R., Raghavan, S., Rangachari, S. S., Lal, S., Algebraic Number Theory, TIFR Mathematical Pamphlet.
2. Artin, M., Algebra, Prentice Hall of India, 1991.
3. Serre, J. P., Local Fields, Graduate Texts in Mathematics, Springer, 1995.
4. Milne, J. S., Algebraic Number Theory, Available online.
5. Ireland, K. and Rosen, M., A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics, Springer, 1982.
6. Stewart, Ian and Tall, David., Algebraic Number Theory and Fermat’s Last Theorem, Taylor & Francis.