Franz Lemmermeyer

# Algebraic Number Theory

### Schedule

```  TU 13:40 - 15:30, SAZ 03
TH 13:40 - 14:30, SAZ 20 ```

### Motivation

A standard course in algebraic number theory discusses the proofs of the main results on integral bases, discriminants, Dedekind rings, class groups, Dirichlet's unit theorem, etc. In this semester, I will instead concentrate on quadratic extensions of the rationals and of the rational function fields and introduce elliptic curves. This will allow us to do a lot of explicit calculations that cannot be done (by hand) for extensions of higher degree.

### Topics

• I. Quadratic Number Fields. Here we will discuss failure of unique factorization, ideal arithmetic, class number computation, and units. Applications: Bachet-Mordell equations y2 = x3+k, some simple 3-descents on elliptic curves like x3+y3 = Az3, possibly the cubic reciprocity law.
• II. Quadratic Function Fields. These are quadratic extensions of the rational function fields Fp[X]; their arithmetic has a lot in common with quadratic number fields.
• III. Arithmetic of Elliptic and Hyperelliptic Curves. These can essentially be identified with quadratic extensions of function fields.

### Homework

Homework is always due one week after hand-out except when stated otherwise. Solutions will be posted after all students have turned their homework in.

### Schedule

• Tu 13.09.05: Euler and the failure of unique factorization
• Th 15.09.05: Algebraic Integers (ring of integers; basic properties of ideals)
• Tu 20.09.05: Ideals and modules
• Th 22.09.05: NO CLASS (moved to Th. 29.09.05)
• Tu 27.09.05: Unique Factorization into prime ideals
• Th 29.09.05: Decomposition law in quadratic number fields.
• Tu 04.10.05: The Pell equation
• Th 06.10.05: Some diophantine equations
• Tu 11.10.05: Principal ideal tests, Computing units. This is the article by Lenstra that I recommended.
• Th 13.10.05: NO CLASS (moved to Th. 27.10)
• Tu 18.10.05: Factoring integers
• Th 20.10.05: Class groups. Here are Chapters 1-6.
• Tu 25.10.05: Quadratic Reciprocity. Here is a review sheet.
• Th 27.10.05: Midterm 1 (solutions)
• Tu 01.11.05: discussion of midterm
• Th 03.11.05: No class
• Tu 08.11.05: Applications to cryptography; here are Chapters 1-6 with an expanded Chapter 6.
• Th 10.11.05: Applications to cryptography
• Tu 15.11.05: Applications to cryptography
• Th 17.11.05: Binary Quadratic Forms.
• Tu 22.11.05: Reduction
• Th 24.11.05: No class (moved to Fr. 16.12)
• Tu 29.11.05: Here are a few practice problems
• Th 01.12.05: Reduction II
• Tu 06.12.05: Gauss Composition
• Th 08.12.05: Gauss Composition II
• Tu 13.12.05: Midterm 2; average 38.3/100. Here are the solutions.
• Th 15.12.05: discussion of midterm, isomorphism
• Fr 16.12.05, 11:00 - 12:00; practice hour
• Mo 19.12.05, 1:30; algebraic number theory; practice for final.
• We 21.12.05, 9:30 - 11:30, SAZ04 Final. Problems Solutions. Average: 53
• Final average of students who turned in homework regularly: 65
• Final average of students who turned in homework irregularly: 42
• Final average of students who did not turn in homework: 27
Do you all see what I see?
• The lecture notes (still details missing in some proofs concerning the group structure on classes of quadratic forms . . .)

### Books

Here are a few books I recommend.
• H. Cohn, Advanced Number Theory. One of the few books with a readable account of quadratic forms.
• D. Flath, Introduction to number theory. Probably the best presentation of the theory of quadratic forms.
• Cox, Primes of the form x2 + ny2: Chapter 1 contains the theory of binary quadratic forms; the other two chapters deal with class field theory and complex multiplication.
• Marcus: Number Fields. My favorite for the theory of general number fields; it has lots of exercises.
• Ireland and Rosen: A classical introduction to modern number theory. An all-time classic.
• Murty and Esmonde: another problem collection.
• Swinnerton-Dyer: A Brief Guide to Algebraic Number Theory: concise, not expensive; if you want to buy a book, this is it.
There are also various lecture notes to be found here.