Math 254A - Number Theory

InstructorPaul Vojta
LecturesMWF 1:10–2:00, Evans 31
Class Number22461
Office883 Evans
Office HoursMondays 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.
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 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.

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.
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
CommentsI tend to follow the book fairly closely, but try to give interesting exercises and examples.


No.DateTitle Download
1August 29 The Discriminant of xn+ax+b dvi pdf
2September 5 Integral bases (corrected version) dvi pdf
3September 17 Tables (Pages 422–425 of Number Theory by Borevich and Shafarevich)
4October 24 Hensel's lemma dvi pdf
5November 14 Absolute values on number fields and function fields dvi pdf
6December 5 𝜁'(0), or the product of the positive integers dvi pdf
7December 5 Synopsis of Class Field Theory dvi pdf

Homework Assignments

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.DueAssignment Notes
18/29 dvi pdf
29/5 dvi pdf Don't do problem 3.
It's been postponed to next week.
39/12 dvi pdf
4Monday 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
610/1 dvi pdf
710/8 dvi pdf 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.
810/15 dvi pdf
910/22 dvi pdf
10 10/29
Do not hand in
dvi pdf
1111/5 dvi pdf
12Wednesday 11/14 dvi pdf
13 Wednesday, 11/28
(rescheduled due to class cancellations)
dvi pdf Don't do #4. The files linked here have been updated to reflect this change.
14 12/3 dvi pdf
15 Thursday, 12/13 5pm dvi pdf

Last updated 19 March 2019