Mathematics 205b

Topics in Number Theory

Algebraic Number Theory I

Winter Quarter 2016

Don Blasius

**Class Time and Location**: MWF 2-2:50 in MS 6201.

Also, 1 further hour weekly for problem solving at time (MWorF), and location, TBA.

**Office hours:** 3-4 MW in office MS 6903

**Texts:
**

Neukirch, Algebraic Number Theory, Springer (comprehensive and contemporary, good problems)

Samuel, Algebraic Theory of Numbers, Dover (extremely clear and concise but too limited in scope)

Lang, Algebraic Number Theory, Springer (a classic, does an enormous amount)

Borevitch and Shafarevitch, Number Theory (great problems)

These are all good texts. I recommend Neukirch and Samuel most for our course.

**Grading**: There will be some graded homework assignments and there will be presentations on topics (with write-ups)at the end of the course. There are no formal exams.
**Themes:** We hope to cover this quarter the content of Chapters I-IV and much of Chapter VII, of Neukirch, assuming for the latter the basic results of class-field theory. Some topics will be sketched or illustrated without full proof. Many examples will be given (some to elliptic curves) and the computer program SAGE (downloadable) will be used to illustrate the theory with examples. In the second quarter we currently plan to develop the theory of heights and study Diophantine problems, using the work done in the first quarter.

A sample of core topics to be covered this quarter is:

- Integrality, Dedekind rings and modules over Dedekind rings
- Traces and Discriminants, Norms
- Geometry of Numbers, esp. Minkowski theory: finiteness of class number, structure of unit group
- Ramification theory
- Cyclotomic fields
- Localization
- Valuation and Completion
- Adeles, Ideles
- Dedekind zeta functions, Hecke L-series, Artin L-series
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**Required Background**

Algebra at Honors undergrad or grad level.