Math 254A - Number Theory

Instructor Paul Vojta

Lectures MWF 5:10–6:00, Dwinelle 234

Class Number 22053

Office 883 Evans

E-Mail: vojta@math.berkeley.edu

Office Hours MWF 1:10–2:00, excluding University holidays and non-instructional days.

Prerequisites Math 250A

Required Text

Neukirch, Algebraic Number Theory, Springer

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

Chapter	Content	Sections
I	Algebraic integers	1–11 and (later) 14
II	Valuations	All but Section 6 and parts of 7, 9, 10
III	Primes, different, discriminant	1, 2, and parts of 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.

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 Wednesdays. 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.

Handouts

No.	Date	Title	Download
1	August 28	The discriminant of xⁿ+ax+b	dvi	pdf
2	September 1	Integral bases	dvi	pdf
4	September 25	Localization	dvi	pdf
5pre	October 27	Valued rings and valued fields (preliminary)	dvi	pdf
5	October 30	Valued rings and valued fields	dvi	pdf
6	November 3	Hensel's lemma	dvi	pdf
7	November 17	An overly long proof of Eisenstein's criterion	dvi	pdf
8	November 20	Absolute values on number fields and function fields	dvi	pdf
9	December 4	Synopsis of class field theory	dvi	pdf
10	December 8	ζ'(0), or the product of the positive integers	dvi	pdf

Homework Assignments

The following policies apply to the homework assignments.

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.)
As an exception to the above, you may not collaborate on problems marked “(nc).”
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).
If you weren't able to do an exercise or part thereof, you may still assume that it holds for subsequent exercises or parts.
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.	Due	Assignment
1	8/30	dvi	pdf
2	9/6	dvi	pdf
3	9/13	dvi	pdf
4	9/20	dvi	pdf
5	9/27	dvi	pdf
6	10/4	dvi	pdf
7	10/11	dvi	pdf
8	10/18	dvi	pdf
9	10/25	dvi	pdf
10	11/1	dvi	pdf
11	11/8	dvi	pdf
12	11/15	dvi	pdf
13	11/28 (Tuesday)	dvi	pdf
14	12/7 (Thursday)	dvi	pdf
15	12/14 (Thursday)	dvi	pdf

Last updated 8 December 2023