| Instructor | Paul Vojta | ||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Lectures | MWF 1:10–2:00, Evans 31 | ||||||||||||||||||
| Class Number | 22461 | ||||||||||||||||||
| Office | 883 Evans | ||||||||||||||||||
| E-Mail: | vojta@math.berkeley.edu | ||||||||||||||||||
| Office Hours | Mondays 11:30–12:30,
Wednesdays 11:10–12; Fridays 11:30–12:30, excluding university holidays and November 21. Please email me to set up a time to meet if you cannot make any of these times. | ||||||||||||||||||
| Prerequisites | Math 250A | ||||||||||||||||||
| Required Text | Neukirch,
Algebraic Number Theory, Springer
The following sections of the book are likely to be covered:
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| 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 |
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, because 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. I will also teach the second half of this course, Math 254B, in Spring 2019. In that course, I plan to cover the more advanced topic of Arakelov theory, including applications to diophantine problems. That course will have Math 256A as an additional prerequisite. | ||||||||||||||||||
| Grading | Grades 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. | ||||||||||||||||||
| RRR Week | Since this is a graduate course, classes will continue through RRR Week. | ||||||||||||||||||
| Homework | Weekly or biweekly, assigned in class, generally due on Mondays | ||||||||||||||||||
| Comments | I tend to follow the book fairly closely, but try to give interesting exercises and examples. |
| No. | Date | Title | Download | ||
|---|---|---|---|---|---|
| 1 | August 29 | The Discriminant of xn+ax+b | dvi | ||
| 2 | September 5 | Integral bases (corrected version) | dvi | ||
| 3 | September 17 | Tables | (Pages 422–425 of Number Theory by Borevich and Shafarevich) | ||
| 4 | October 24 | Hensel's lemma | dvi | ||
| 5 | November 14 | Absolute values on number fields and function fields | dvi | ||
| 6 | December 5 | 𝜁'(0), or the product of the positive integers | dvi | ||
| 7 | December 5 | Synopsis of Class Field Theory | dvi | ||
In these assignments, 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 handed in in class.
Solutions are posted on bCourses after each assignment has been graded.
| No. | Due | Assignment | Notes | |
|---|---|---|---|---|
| 1 | 8/29 | dvi | ||
| 2 | 9/5 | dvi | Don't do problem 3. It's been postponed to next week. | |
| 3 | 9/12 | dvi | ||
| 4 | Monday 9/17 | p. 27: 2, 3; p. 34: 1, 3 |
Don't forget to prove the last sentence of #2 on page 27. | |
| 5 | 9/24 Do not hand in |
p. 15: 5; p. 38: 6, 7; p. 43: 1,6 |
||
| 6 | 10/1 | dvi | ||
| 7 | 10/8 | dvi | Don't do Problems 3 and 4. Instead do Exercise 7 on page 52. The files linked here have been updated to reflect this change. | |
| 8 | 10/15 | dvi | ||
| 9 | 10/22 | dvi | ||
| 10 | 10/29 Do not hand in |
dvi | ||
| 11 | 11/5 | dvi | ||
| 12 | Wednesday 11/14 | dvi | ||
| 13 | Wednesday, 11/28 (rescheduled due to class cancellations) |
dvi | Don't do #4. The files linked here have been updated to reflect this change. | |
| 14 | 12/3 | dvi | ||
| 15 | Thursday, 12/13 5pm | dvi | ||
Last updated 19 March 2019