Subject Code: MA7L027	Subject Name: Algebraic Number Theory	L-T-P: 3-0-0	Credit: 3
Pre-requisite(s): Algebra MA5L006
Syllabus Basic Theory of Field Extensions, Algebraic Extensions, Splitting Fields, Algebraic Closure, Separable and Inseparable Extensions, Galois groups; Quadratic Fields, Cyclotomic Fields, The Ring of Integers, Embeddings in C; Trace and the Norm, Existence of an Integral basis, The discriminant of an n-tuples, Ideals in Ring of Integers The additive structure of a Number Ring; Integral Closure, Characterizing Dedekind Domains, Fractional Ideals and Unique Factorizations, Dedekind’s Theorem, Factorization in the Ring of Integers; Splitting of primes in Extensions; Finiteness of Ideal Class Group, Diophantine Equations, Exponents of Ideal Class Groups; The Dirichlet’s Unit Theorem, Units in Real Quadratic Fields; The Riemann and Dedekind Zeta functions, Zeta functions of Quadratic Fields, Dirichlet’s L-functions, Primes in Arithmetic progressions, The Class Number Formula, Example: The Quadratic Case. Text Books: Daniel A. Marcus, Number Fields, Springer, Second Edition, 2018. I. Stewart and D. Tall, Algebraic Number Theory and Fermat’s Last Theorem, Fourth Edition, Peters, 2016. Reference Books: Jurgen Neukirch, Algebraic Number theory, Springer, 1999. J J. P. Serre, Local Fields, Graduate Texts in Mathematics 67, Springer, 1979. M. R. Murty, and J. Esmonde, Problems in Algebraic Number Theory, Graduate Studies in Mathematics, 190, Springer-Verlag New York, 2005.

Subject Code: MA7L027

Subject Name: Algebraic Number Theory

L-T-P: 3-0-0

Credit: 3

Pre-requisite(s): Algebra MA5L006

Syllabus
Basic Theory of Field Extensions, Algebraic Extensions, Splitting Fields, Algebraic Closure, Separable and Inseparable Extensions, Galois groups; Quadratic Fields, Cyclotomic Fields, The Ring of Integers, Embeddings in C; Trace and the Norm, Existence of an Integral basis, The discriminant of an n-tuples, Ideals in Ring of Integers The additive structure of a Number Ring; Integral Closure, Characterizing Dedekind Domains, Fractional Ideals and Unique Factorizations, Dedekind’s Theorem, Factorization in the Ring of Integers; Splitting of primes in Extensions; Finiteness of Ideal Class Group, Diophantine Equations, Exponents of Ideal Class Groups; The Dirichlet’s Unit Theorem, Units in Real Quadratic Fields; The Riemann and Dedekind Zeta functions, Zeta functions of Quadratic Fields, Dirichlet’s L-functions, Primes in Arithmetic progressions, The Class Number Formula, Example: The Quadratic Case.

Text Books:

Daniel A. Marcus, Number Fields, Springer, Second Edition, 2018.
I. Stewart and D. Tall, Algebraic Number Theory and Fermat’s Last Theorem, Fourth Edition, Peters, 2016.

Reference Books:

Jurgen Neukirch, Algebraic Number theory, Springer, 1999.
J J. P. Serre, Local Fields, Graduate Texts in Mathematics 67, Springer, 1979.
M. R. Murty, and J. Esmonde, Problems in Algebraic Number Theory, Graduate Studies in Mathematics, 190, Springer-Verlag New York, 2005.