Instructor | Paul Vojta | ||||||||||||||||||
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Lectures | MWF 1:10–2:00, Giannini 201 | ||||||||||||||||||
Class Number | 22319 | ||||||||||||||||||
Office | 883 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 | ||||||||||||||||||
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 (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. | ||||||||||||||||||
Grading | Grades 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. | ||||||||||||||||||
Comments | I tend to follow the book fairly closely, but try to give interesting exercises and examples. |
No. | Date | Title | Download | ||
---|---|---|---|---|---|
1 | August 30 | The Discriminant of xn+ax+b | dvi | ||
2 | September 5, 8 | Integral bases (revised September 8) | dvi | ||
3 | September 20 | Tables | (Pages 422–425 of Number Theory by Borevich and Shafarevich) | ||
4 | November 1 | Hensel's lemma | dvi |
The following policies apply to the homework assignments.
Homework assignments should be submitted on bCourses, unless otherwise specified.
Solutions are posted on bCourses after each assignment has been graded.
No. | Due | Assignment | Notes | |
---|---|---|---|---|
1 | 9/1 | dvi | ||
2 | 9/8 | dvi | ||
3 | 9/15 | dvi | ||
4 | 9/22 | dvi | ||
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. | |
6 | 10/6 | dvi | ||
7 | 10/13 | dvi | ||
8 | 10/20 | dvi | ||
9 | 10/27 | dvi | ||
10 | 11/3 | dvi | ||
11 | 11/10 | dvi | ||
12 | 11/17 | dvi | ||
13 | 11/30 (Tuesday) |
dvi | ||
14 | 12/9 (Thursday) |
dvi | ||
15 | 12/16 (Thursday) |
dvi |
Last updated 17 December 2021