Course Syllabus: Click here.
Weekly Schedule: Announcements about homework, quizzes, exams, special lectures, etc. will be posted here.
1. Week of January 2nd.
Lecture 1 (04/01/17): Unique factorization in Z and applications: Euler product for Riemann zeta fn., sum of reciprocals of primes diverges.
Lecture 2 (05/01/17): Euclidean domain, UFD, PID.
Lecture 3 (06/01/11): Gaussian integers: Z[i]. Units; primes; factorization of rational primes in Z[i].
Homework 1: Click here.
2. Week of January 9th.
Lecture 4 (11/01/17): Fermat's little theorem, Euler's theorem, Wilson's theorem via group theory; infinitely many primes of the form 4k+1.
Lecture 5 (12/01/17): Linear congruences; Chinese reminder theorem, and it's ring-theoretic version; weak approximation for Z and finite set of primes.
Lecture 6 (13/01/11): Structure of the group of units in Z/nZ; primitive roots.
Homework 2: Click here.
3. Week of January 16th.
Lecture 7 (18/01/17): Quadratic residues and non-residues; finding square roots mod n and mod p-powers; Legendre symbol and properties.
Lecture 8 (19/01/17): Proof of Quadratic reciprocity via Gauss sums.
Lecture 9 (20/01/11): A primer on finite fields.
Homework 3: Click here.
4. Week of January 23rd.
Lecture 10 (25/01/17): Integral extensions; ring of integers in a number field.
(No lecture on 26/01/17: Republic Day.)
Lecture 11 (27/01/17): Trace and norm map; Galois theoretic interpretation of these maps: averaging over conjugates.
Homework 4: Click here.
5. Week of January 30th.
Lecture 12 (01/02/17): Review of non degenerate bilinear forms.
Lecture 13 (02/02/17): Discriminant of a basis; non degeneracy of the trace-form for a separable extension.
No lecture on 03/02/17, as the instructor is busy with Board of Governors meeting.
No homework set.
6. Week of February 6th.
Lecture 14 (08/02/17): Discriminant of a number field.
Lecture 15 (09/02/17): Ring of integers of a number field is a Dedekind domain.
Lecture 16 (10/02/17): General properties of a Dedekind domain.
Homework 5: Click here.
7. Week of February 13th.
Lecture 17 (15/02/17): A review of Dedekind domains.
Lecture 18 (16/02/17): n = e_1f_1+…+e_gf_g.
Lecture 19 (17/02/17): The above formula, continued.
8. Week of February 20th.
Midterm exam on February 24th from 10:00 am to 12 noon.
9. Week of February 27th.
Lecture 20 (01/03/17): Discuss midterm exam.
Lecture 21 (02/03/17): Discuss projects: Riemann zeta function, Dirichlet L-function, Dedekind zeta function, Gauss and Jacobi sums, Cyclotomic fields,
Dirichlet's class number formula, Fermat's last theorem for regular primes, Pell's equations and units in a real quadratic field.
(03/03/17): Class cancelled.
10. Week of March 6th.
Lecture 22 (08/03/17): Splitting behavior of a prime p via splitting behavior of minimal polynomial mod p.
Lecture 23 (09/03/17): Splitting behavior of a prime p, continued…
Lecture 24 (10/03/17): Hilbert's ramification theory; Galois group acts transitively on the primes above a given prime below.
Homework 6: Click here.
11. Week of March 13th.
Lecture 25 (15/03/17): Decomposition group and decomposition field.
Lecture 26 (16/03/17): Inertia group and inertial field.
Lecture 27 (17/03/17): Example: primes factoring in the field obtained by adjoining 2^{1/3} and cube root of unity.
Homework 7: Click here.
12. Week of March 20th.
Lecture 28 (22/03/17): Motivation on adèles and idèles.
Lecture 29 (23/03/17): Get started on Z_p and Q_p. Hensel's definition of Z_p as an inverse limit of Z/p^n.
(17/03/17): Lecture cancelled.
No homework this week.
13. Week of March 27th.
Lecture 30 (29/03/17): Q_p as the completion of Q with respect to the p-adic valuation; and Z_p is the unit ball in Q_p.
Lecture 31 (30/03/17): Equivalence of the two definitions of Q_p and Z_p.
Lecture 32 (31/03/17): Valuations.
Lecture 33/34 (01/04/17): Valuations and localizations. (Special 2-hour lecture starting 10:30 a.m. in Madhava Hall.)
Homework 8: Click here.
14. Week of April 3rd.
Lecture 35 (05/04/17): Completions and local fields.
Lecture 36 (06/04/17): Normalized valuations and the produce formula.
Lecture 37 (07/04/17): The definition of adèle ring A_F, the idèle group I_F. The adelic norm.
Homework 9: Click here.
15. Week of April 10th.
Lecture 38 (12/04/17): Compactness of A_F/F.
Lecture 39 (13/04/17): Compactness of I_F^1/F^*.
Lecture 40 (14/04/17): Finiteness of class number. Dirichlet's unit theorem.
Special Session (15/04/17): Project presentations. (Starts 10:00 a.m. in Madhava Hall.)
Homework 10: Click here.
16. Week of April 17th.
Lecture 41 (19/04/17): Concluding lecture.
17. Week of April 24th.
Final Exam on Tuesday, 25th April, from 10:00 am to 12:00 noon, in LHC 201.