Math 254A - Number Theory

InstructorPaul Vojta
LecturesMWF 1:10–2:00, Giannini 201
Class Number 22319
Office883 Evans
E-Mail:vojta@math.berkeley.edu
Office Hours MWF 12:10–1:00 and MWF 3:10–4:00, excluding University holidays.
During Finals Week: TTh 11:10–12
PrerequisitesMath 250A
Required TextNeukirch, Algebraic Number Theory, Springer

The following sections of the book are likely to be covered:

Chapter Content Sections
I Algebraic integers 1–12
II Valuations All but Section 6 and parts of 7, 9, 10
III Primes, different, discriminant 1–3
VII Zeta functions and L-series A small subset
VI Class field theory 1, 2, and a few other parts
Catalog Description (of Math 254AB) 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 This is the standard first-year graduate course on number theory. In the fall semester the course will cover the basics of number theory over a Dedekind domain: completions, fractional ideals, ideles and adeles, etc., as in the catalog description or the first three chapters of the textbook. Basically, the idea is to study finite algebraic extensions of Z or Q and determine which properties still hold in this more general setting. Often the structure of a system of diophantine equations over Z or Q is more apparent after extending the field of definition.

The course will also include some introductory material on analytic number theory and class field theory.

GradingGrades will be based on homework assignments. There will be no final exam, but the last problem set will be due sometime during the week of final exams.
RRR Week Since this is a graduate course, classes may continue through RRR Week.
Homework Weekly or biweekly, assigned in class, generally due on the same day each week (TBD). Assignments are given below. Solutions will be posted on bCourses.
CommentsI tend to follow the book fairly closely, but try to give interesting exercises and examples.

Handouts

No.DateTitle Download
1August 30 The Discriminant of xn+ax+b dvi pdf
2September 5, 8 Integral bases (revised September 8) dvi pdf
3September 20 Tables (Pages 422–425 of Number Theory by Borevich and Shafarevich)
4November 1 Hensel's lemma dvi pdf

Homework Assignments

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, for each problem you must list the names of students with whom you have collaborated on that particular problem. (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. 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).
  4. If you weren't able to do an exercise or part thereof, you may still assume that it holds for subsequent exercises or parts.
  5. Likewise, you may use earlier exercises from the book without proving them. (For problems not from the book, use common sense or ask me.)

Homework assignments should be submitted on bCourses, unless otherwise specified.

Solutions are posted on bCourses after each assignment has been graded.

No.DueAssignment Notes
19/1 dvi pdf
29/8 dvi pdf
39/15 dvi pdf
49/22 dvi pdf
5 9/29
Do not hand in
p. 15: 5;
p. 38: 6, 7;
p. 43: 1,6
For Exercise 5 on page 15: avoid a tedious repetition of cases.
610/6 dvi pdf
710/13 dvi pdf
810/20 dvi pdf
910/27 dvi pdf
1011/3 dvi pdf
1111/10 dvi pdf
1211/17 dvi pdf
13 11/30
(Tuesday)
dvi pdf
14 12/9
(Thursday)
dvi pdf
15 12/16
(Thursday)
dvi pdf

Last updated 17 December 2021