- 1. Fundamentals: the construction of natural numbers, integers, and rational numbers from the Peano axioms.
- 2. Basics: divisibility, unique factorization, congruences, primes, . . .
- 3. Usual Suspects: infinitude of primes, theorems of Fermat-Euler, two squares, quadratic reciprocity, . . .
- 4. Simple applications to cryptography, codes, primality tests, . . .
- 5. p-adic numbers
- 6. Quadratic forms

- Mo 07.02.05 problems solutions
- Mo 14.02.05 problems solutions
- Mo 28.02.05 problems solutions
- Mo 07.03.05 problems solutions
- We 16.03.05 problems solutions
- Mo 18.04.05 problems solutions
- Mo 02.05.05 problems you may hand in the homework on Wednesday, but don't look at the solutions!

- Rose: A course in number theory
- Hardy and Wright: An introduction to the theory of numbers
- Ireland and Rosen: A classical introduction to modern number theory
- Frey: Elementare Zahlentheorie

- We 02.02.05: natural numbers
- Mo 07.02.05: integers and rational numbers
- We 09.02.05: Unique Factorization
- Mo 14.02.05: Euclidean Algorithm; Diophantine equations. Here is a proof of Lamé's Theorem that Fibonacci numbers produce the longest chains in the Euclidean algorithm.
- We 16.02.05: *Extra class* More diophantine problems
- Mo 21.02.05: FLT for exponent 4; Two squares
- We 23.02.05: no class
- Mo 28.02.05: Quadratic Reciprocity (updated 07/03)
- We 02.03.05: Quadratic Reciprocity
- Mo 07.03.05: Euler-Fermat
- We 09.03.05: *Extra class* Chebyshev
- Mo 14.03.05: the RSA cryptosystem
- We 16.03.05: Euclidean rings (updated 20/04)
- Mo 21.03.05: Gaussian Integers (updated 20/04). Check out Keith Conrad's notes on Gaussian integers. He also has notes on quadratic residues at the bottom of his page.
- We 23.03.05: *Extra class* : special cases of Dirichlet's theorem on primes in arithmetic progressions.
- Mo 28.03.05: Residue classes in Z[i]
- We 30.03.05: Midterm 1 (review sheet)
- Mo 04.04.05: The Rational Function Field (updated 20/04)
- We 06.04.05: no class
- Mo 11.04.05: spring break
- We 13.04.05: spring break
- Mo 18.04.05: Legendre symbols in Z[i] and F
_{p}[X] - We 20.04.05: Quadratic Reciprocity in Z[i]; primitive roots
- Mo 25.04.05: Quadratic Reciprocity in Z via Gauss sums;
Quadratic Reciprocity
in F
_{p}[X] - We 27.04.05: Valuations, real and p-adic numbers; check out the links to the articles by Goldstein, Gouvea and Madore on this page
- Mo 02.05.05: counterexamples to the local-global principle. I also mentioned a problem I cannot solve: let M and N be monoids of natural numbers, and assume that M and N have unique factorization. Does this imply that the intersection of M and N also has unique factorization?
- We 04.05.05: no class
- Mo 09.05.05: discussion of homework problems
- We 11.05.05: CHANGE OF SCHEDULE: midterm postponed, no class
- Th 12.05.05: 18:00 (SAZ 04) Midterm 2; you should be able to do
arithmetic in the rings Z[i], F
_{p}[X], and Z_{p}(chapters 10-13). Here are a few review problems (UPDATED 11.05.05, 19:30). solutions (average: 63) - Mo 13.05.05:
- We 15.05.05: review
- Midterm 2 make-up problems solutions
- Final (here are some test problems.

If you want to read something on your own, see whether you can find a copy of Buchmann's book in one of the libraries around here. You may also look at lecture notes on the web, such as those listed here.