Franz Lemmermeyer
Elementary Number Theory
Topics
- 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
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.
Books
Here are a few books I recommend
- 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
It is sufficient to be familiar with the lecture notes on this page.
This remark should not keep you from looking at other books, however.
Schedule
- 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 Fp[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 Fp[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], Fp[X],
and Zp (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.
I've been asked about cryptography by some. I will probably
offer an unofficial course on crypto and coding next semester.
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.