Elementary Number Theory
Here's a link
to last year's course.
- 1. Basics: divisibility, unique factorization, congruences, primes, . . .
- 2. Usual Suspects: infinitude of primes, theorems of Fermat-Euler,
two squares, quadratic reciprocity, . . .
- 3. Simple applications to cryptography, codes, primality tests, . . .
- 4. Rational Function Fields
- 5. Quadratic forms
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.
Here are a few books I recommend
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.
- 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
Additional Lecture notes
(topics in n.th.),
here (everything is in postscript),
- 02.02.06 Monoids, unique factorization (sorry for hurrying through
the last proof; I'll do it again next time).
- 07.02.06 residue classes
- 09.02.06 greatest common divisors, Euclidean algorithm
- 14.02.06 Euclidean algorithm, Pythagorean triples
- 16.02.06 Fermat's Theorem on the diophantine equation
x4 + y4 = z4.
- 21.02.06 Fermat's Little Theorem; Z/pZ is a field.
- 28.02.06 Residue class rings; Pollard's p-1 method.
you can find the record factors found with this method.
- 02.03.06 NO CLASS: moved to 04.03.06
- Sat. 04.03.06, 10:30-12:30
Midterm 1 in BZ08
(Computer Center Building). We will cover the material up to 21.02.06.
Average: 67/100. Here are a few similar
for those who need more practice.
- 07.03.06 Discussion of Midterm 1. Theorem of Euler-Fermat
- 09.03.06 Chinese Remainder Theorem
- 14.03.06 Chinese Remainder Theorem (applications);
order of elements; Mersenne and Fermat numbers
- 16.03.06 primitive roots
- 21.03.06 primitive roots; quadratic reciprocity
- 23.03.06 NO CLASS: moved to 04.03
- 04.04.06 quadratic residues; Legendre symbol.
- 06.04.06 the quadratic reciprocity law
- 11.04.06 Quadratic reciprocity; Jacobi symbol.
- 13.04.06 discussion of homework; binary quadratic forms
- 18.04.06 homework; review; quadratic forms
- 20.04.06 Midterm 2,
SBZ03, 5:40 - 7:40 pm (no class at 9:40).
We will cover phi function, Euler-Fermat, RSA, Chinese Remainder
Theorem, order of residue classes, primitive roots, quadratic
reciprocity. Average: 71/100.
- 25.04.06 Reduction.
- 27.04.06 Class numbers
- 02.05.06 Class numbers
- 04.05.06 Quadratic number fields
- 09.05.06 Unique Factorization and class number 1
- 11.05.06 Composition
- 16.05.06 Composition; review
- 17.05.06 Final 12.15, SAZ04.
- Here is the complete set of notes.
- Here are a few problems for the
last two chapters. Here are the
for a make up midterm 2, along with partial
- We 18.05, 10:30 - 12:00 I'll be in my office.
- Next week: Oberwolfach. Mathematicians at
- Mo 29.05, 13:30 make up