his is a course webpage of MT3224; an undergraduate course on algebraic number theory. Please read the course template for text-books and other important information about the course. This is a 4 credit course.
Venue-LHC 106
Mid-Semester Exams-30%
End-Semester Exams-30%.
3. Class tests (4):-10%+10%+10%+10%.
Number fields-Marcus
Algebraic number theory (Neukirch).
Algebraic theory of numbers-Pierre Samuel
Problems in algebraic number theory: Esmonde and Ram Murty
30/01/2023: Class test 1.
13/02/2023:Class test 2.
Galois theory
Class timing:
Mon- 10 am
Tue-10 am
Wednesday-10 am
Week 1
What is algebraic number theory?
Recall the definition of Euclidean domain, UFD, PID (Dummit and Foot and Pierre Samuel)
Week 2
Ring of integers of number fields,
Integral elements. Definition, basic properties and examples of Integral elements.
Week 3
Trace induces a non-degnerate bilinear form, O_K is free (again), other properties of O_K.
:Discriminants:- Definition and basic properties of the Discriminants.
Week 4
Finding integral basis of O_K
Integral basis of cyclotomic fields
Week 5
Dedekind domains:- Definition, examples and counter examples of Dedekind domains.
Fractional ideals and it's unique factorization.
Week 8
Recall Chinese Remainder theorem. The formula n=\sum_i^r e_i f_i
Recall the wonderful formula n=\sum_i^r e_i f_i; Towards Dedekinds' theorem
Tutorial
Weak 9
How to find e_i, f_i, r in the wonderful formula as above.
Week 10
Kummer's theorem and how to find factorization.
Week 11
Primes ramify if and only if it divide the discriminant.
Week 12
Fractional ideals and finiteness of class groups (Marcus's proof).
Minkowski's Lattice point theorem
Week 11
How to calculate class groups.
Week 12:
Dirichlet's unit theorem.
Week 13
Decomposition groups and inertia groups.