Math 254B - Number Theory

InstructorPaul Vojta
LecturesMWF 10–11, Evans 736
Class Number 22425
Office883 Evans
E-Mail:vojta@math.berkeley.edu
Office HoursMWF 11:30–12:30
excluding non-instructional days.
Please email me to set up a time to meet if you cannot make any of these times.
GSI Office HoursThe GSI for Math 254B is Rahul Dalal. He will hold office hours on MTuF 2–3 (omitting Wednesdays) and Thursdays 10–11 and 1–3, in 1041 Evans.
PrerequisitesMath 254A
Required TextJ. Neukirch, Algebraic number theory, Springer, 2010
Catalog Description Valuations, units, and ideals in number fields, ramification theory, quadratic and cyclotomic fields, topics from class field theory, zeta-functions and L-series, distribution of primes, modular forms, quadratic forms, diophantine equations, p-adic analysis, and transcendental numbers.
Syllabus I plan to cover Class Field Theory, as given in Chapters IV–VI of the textbook.
GradingGrades will be based on homework assignments, and possibly in-class quizzes. There will be no final exam, but the last problem set will be due sometime during the week of final exams.
Homework Weekly or biweekly, assigned in class

Handouts

No.DateTitle Download
1January 27 Synopsis of class field theory dvi pdf

Homework Assignments

Solutions will be posted on bCourses after each assignment has been graded.

To submit your assignments, use bCourses. I have set it to accept only pdf files.

No.DueAssignment Notes
11/31 pp. 367–368: 3, 11, 12, 13. For #3, If1 is defined in the proof of (1.11), and the completion of ℤ is defined on page 272 (Example 4).
22/7 p. 368: 14; p. 265: 4, 5; p. 274: 3(nc). For the second problem, note that profinite group is defined on page 264.
32/14 dvi pdf Exercise 4 on page 290 is no longer assigned.
42/28 dvi pdf This replaces the earlier Homework 4, originally due on Feb. 21
53/6 p. 298–299: 1, 3 Hint for #1: Use the method of the proof of Proposition (5.5), with one of the factors σi replaced by an element of G(L-tilde/K-tilde).
Also for #1: There's a sign error. Show instead that rL/K(σ)=N(y)-1.
63/20 dvi pdf Correction (3/14): For the last question, assume that G is finite cyclic.
74/3 dvi pdf

The following policies apply to the homework assignments.

  1. Discussion and the exchange of ideas are essential to doing academic work. For assignments in this course, you are encouraged to consult with your classmates as you work on problem sets. However, after discussions with peers, make sure that you can work through the problem yourself and ensure that any answers you submit for evaluation are the result of your own efforts. In addition, you must cite any books, articles, websites, lectures, etc. that have helped you with your work using appropriate citation practices (other than class lectures and corresponding parts of the textbook). Similarly, you must list the names of students with whom you have collaborated on problem sets. (This paragraph was adapted from http://oue.fas.harvard.edu/icb/icb.do?keyword=k18059&pageid=icb.page498261, accessed on 28 January 2013.)
  2. As an exception to the above, you may not collaborate on problems marked “(nc).”
  3. For problems from the book, you may use without proof any lower-numbered exercises from the same section or any exercises from sections covered earlier. For other exercises, you may use without proof any exercises from earlier sections in the book; for the current section(s), ask. For any exercises, if you weren't able to do an exercise or part of an exercise, you may still assume that it holds for subsequent exercises or parts.
  4. Credit may be reduced if you look up the answer in some other source (as opposed to using the material from elsewhere to aid in your understanding), or if you look ahead in the textbook.

Last updated 27 March 2020