Algebraic Number Theory Spring 2017

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.