Math 105: Algebraic number theory

Fall 2014


Course Info:



[PDF] Syllabus

This course will be a graduate-level introduction to algebraic number theory, in which we will cover the fundamentals of the subject. Topics may include: rings of integers, Dedekind domains, factorization of prime ideals, Galois theory in number fields, geometry of numbers and Minkowski's theorem, finiteness of the class number, Dirichlet's unit theorem, selected topics from analytic number theory, quadratic and cyclotomic fields, localization and local rings, valuations (i.e. p-adic) and completions, an introduction to class field theory, application to Diophantine equations, and other topics as time permits.



The homework assignments will be assigned on a weekly basis and will be posted below. Homework is required for undergraduates and graduate students in years 1 or 2; it is optional but strongly encouraged for anyone else. In general, it is due in one week, but late homework will be accepted.

Cooperation on homework is permitted (and encouraged), but if you work together, do not take any paper away with you--in other words, you can share your thoughts (say on a blackboard), but you have to walk away with only your understanding. In particular, you must write the solution up on your own.

Plagiarism, collusion, or other violations of the Academic Honor Principle, after consultation, will be referred to the The Committee on Standards.

[PDF] Homework Submission Guidlines

115 Sep(M)Introduction, I.1 (Gaussian Integers)
217 Sep(W)I.2 (Integrality)
319 Sep(F)I.2
422 Sep(M)I.2
524 Sep(W)I.3 (Ideals)I.1.3, I.2.3, I.2.6
626 Sep(F)I.3
729 Sep(M)I.3, I.4 (Lattices)
830 Sep(T)I.4 (Lattices)
91 Oct(W)I.5 (Minkowski Theory)
103 Oct(F)I.5, I.6 (The Class Number)I.3.1, I.3.5, I.3.8, I.3.9
116 Oct(M)I.6
127 Oct(T)I.6
138 Oct(W)I.7 (Dirichlet's Unit Theorem)
1410 Oct(F)I.7, I.8 (Extensions of Dedekind Domains)I.4.1, I.5.3 (may use I.5.2 without proof), I.6.3
1513 Oct(M)I.8
1615 Oct(W)I.8, I.9 (Hilbert's Ramification Theory)
1717 Oct(F)I.9, I.10 (Cyclotomic Fields)
1820 Oct(M)I.10I.7.4, I.7.5, I.8.2, I.8.8
1921 Oct(T)I.10
2022 Oct(W)I.11 (Localization)
-24 Oct(F)No class (JV at NYU)
2127 Oct(M)I.11, II.1 (The p-adics)
2229 Oct(W)II.2 (The p-adics)I.9.1, I.9.2 (typo: q=N(pp)), I.10.2 (assume I.10.1), I.11.6, I.11.A
2331 Oct(F)II.2
243 Nov(M)II.2, II.3 (Valuations)
254 Nov(T)II.3
265 Nov(W)II.4 (Completions)
277 Nov(F)II.5 (Local Fields)II.1.1, II.1.5, II.2.4, II.3.3
-10 Nov(M)No class (JV in Montreal)
-12 Nov(W)No class (JV in Montreal)
2814 Nov(F)
2917 Nov(M)