# Quadratic Number Fields

## Summer term 2016

Lecturer: Tommy Hofmann

Lecture: Monday, 11:45-1:15, in 48-438 (first lecture is on April 18)

Tutorial: Friday, 11:45-1:15, in 48-438 (every two weeks, starting on May 2)

If you have any problems, questions or comments, feel free to contact me via email or drop by my office.

Due to public holiday there will be no lecture on May 16.

### Contents

The investigation of Diophantine equations is a classical (and ongoing) problem in number theory. A typical example is the determination of all Pythagorean triple, that is, all triples (x,y,z) \in \mathbf Z^3 with x^3 + y^3 = z^3, which can be done using elementary techniques. Another example would be the determination of all tuples (x,y) \in \mathbf Z^2 satisfying the equation y^2 = x^3 - 2, a problem studied 300 years ago by Fermat and Euler. Although we reduced the number of variables by 1, the problem gets much harder and elementary techniques cannot help us. Now Euler had the ingenious idea of writing the equation in the following form

x^3 = y^2 + 2 = (y - \sqrt{-2})(y + \sqrt{-2})

and then working in the ring \mathbf Z[\sqrt{-2}] = { a + b \sqrt{-2} | a,b \in \mathbf Z}, the ring of integers of the quadratic field \mathbf Q(\sqrt{-2}). Note that the investigation of y^2 = x^3 - 2, a problem involving only elements of \mathbf Z, has lead us naturally to much more involved objects. In this particular case the new objects are subrings of the quadratic number fields \mathbf Q(\sqrt{d}), d \in \mathbf Z, and the investigation of these rings will be the topic of this course. Of particular interest will be the failure of unique factorization and ways to quantify and overcome this via the ideal class group. To do this, we will employ algebraic, geometric and analytic techniques.

### Credit Points and Problem Sets

There will be problem sets every two weeks, each of which typically contains four problems. To participate in the tutorial, please register at URM till April 22. To obtain a “Schein”, the following criteria have to be met:

• Attending the tutorial (at most one tutorial may be missed).
• Scoring at least 50% of the points.

For bachelor students, the “Schein” is a prerequisite for the oral exam, where you then can get 4.5 credit points. On the other hand, master students can always register for the oral exam (this is independent of the “Schein”). Without the “Schein” passing the oral exam will give only 3 credit points, while passing the oral exam with the “Schein” will give 4.5 credit points.

### Prerequisites

Basic knowledge of groups, rings and fields, which is taught for example in the course “Algebraische Strukturen”.

### Literature

There will be weekly updated lecture notes here. The material of the course is (partly) covered in the following books:
• Mak Trifković, Algebraic Theory of Quadratic Numbers, Springer, 2013
• Michael Artin, Algebra, Prentice-Hall, 1991 (Chapter 11)