Schedule for Algebraic Number Theory 2007 Course
This is the plan for this course. We will not do everything in each listed chapter in class, and students will be expected to read in the book everything we skip in class. The prerequisites are fairly modest and the material is extremely interesting and useful and foundational to other mathematics.
Week 1: * (sept 26) Introduction to algebraic number theory * (sept 28) Finitely generated abelian groups Week 2: LECTURER THIS WEEK: Trevor Arnold * (Oct 1) Noetherian rings (part 1) * (Oct 3) The Hilbert Basis theorem (with proof) * (Oct 5) Algebraic integers; norm and trace Week 3: * (Oct 8) Dedekind domains and the ring of integers * (Oct 10) Introduction to Sage * (Oct 12) The group of fractional ideals in a Dedekind domain Week 4: * (Oct 15) Unique factorization of ideals; examples * (Oct 17) Factoring rational primes in number fields: special algorithm * (Oct 19) Factoring rational primes in number fields: general algorithm Week 5: * (Oct 22) No Class * (Oct 24) The generalized Chinese remainder theorem * (Oct 26) GUEST LECTURER (Koopa Koo) -- Cyclotomic fields Week 6: * (Oct 29) Discriminants and ramification * (Oct 31) Norms of ideals; start of class groups * (Nov 2) [receive midterm] Class groups of number fields: definition and examples Week 7: * (Nov 5) [midterm due] Proof of finiteness of the class group: part 1 * (Nov 7) Proof of finiteness of the class group: part 2 * (Nov 9) LECTURER (T Arnold) -- Unit groups and Dirichlet's Unit Theorem Week 8: * (Nov 12) No class. * (Nov 14) LECTURER (T Arnold) -- Proof of Dirichlet's Unit Theorem * (Nov 16) Decomposition of primes in Galois extensions: e, f, and n Week 9: * (Nov 19) The Decomposition and Inertia Groups * (Nov 21) Frobenius elements and Galois representations * xx (Nov 23) Thanksgiving xx Week 10: * (Nov 26) LLL lattice basis reduction * (Nov 28) A glimpse of Zeta functions of number fields and the class number formula * (Nov 30) Project Discussions, questions, etc. Week 11: * (Dec 3) The Riemann Hypothesis * (Dec 5) The Birch and Swinnerton-Dyer conjecture, part 1 * (Dec 7) The Birch and Swinnerton-Dyer conjecture, part 2 PROJECTS due Dec 7.