Elements of Elementary and Algebraic Number Theory (Reading course) SS 2012

Tuesday, 14:00, room B04, Kampus Kirchberg

Organizational matters (to be supplemented)

About the course: Every week there will be a lecture on a given subject prepared by a student followed by a discussion (questions, remarks, etc.).
The first lecture on February 21, 2012 will be given by me.



Requirements to the speakers.



The speaker is supposed to give a lecture on a given subject. The duration of the lecture has to be between 55 and 65 minutes. The speaker is supposed to provide his or her lecture with a series of at least 3 exercises which are to be in direct connection with the substance of the lecture. Written notes of the presentation as well as the exercise sheet with solutions are to be submitted on the day of the presentation.



Requirements to the participants.



Within one week after the lecture every participant has to submit the solutions of the exercises proposed in that lecture. The participant is also supposed to be able to explain and present his or her solution to the audience.



Evaluation.


For successful completion of the course every student has
to successfully present the required number of lectures,
to actively participate in the discussions (in particular it implies ones presence in the meetings!),
to reasonably treat at least 70% of the proposed exercises and to present some of the solutions to the audience.
A more or less complete explanation of the evaluation system can be found here .


Meetings

Lectures: Exercises:
Lecture 1, Tuesday, February 21, 2012, Speaker: Oleksandr Iena
Topic: Greatest common divisor for integer numbers and linear diophantine eqiations.
Exercises from Lecture 1 ps-file , pdf-file
Lecture 2, Tuesday, February 28, 2012, Speaker: Kathleen Franzen
Topic: Euclidean and factorial rings.
Exercises from Lecture 2 ps-file , pdf-file
Lecture 3, Tuesday, March 6, 2012, Speaker: Ricardo De Sousa
Topic: Continued fractions.
Exercises from Lecture 3 ps-file , pdf-file
Lecture 4, Tuesday, March 13, 2012, Speaker: Mike Schomer
Topic: Quadratic residues.
Exercises from Lecture 4 ps-file , pdf-file
Tuesday, March 20, 2012
NO LECTURE.
Exercises from previous lectures ps-file , pdf-file
Lecture 5, Tuesday, March 27, 2012, Speaker: Amide Dervishi
Topic: Euler's theorem and RSA algorithm.
Exercises from Lecture 5 ps-file , pdf-file
Lecture 6, Tuesday, April 10, 2012, Speaker: Mike Schomer
Topic: Quadratic residues (II). Legendre and Jacobi symbols.
Exercises from Lecture 6 ps-file , pdf-file
Lecture 7, Tuesday, April 17, 2012, Speaker: Kathleen Franzen
Topic: Primality tests
Exercises from Lecture 7 ps-file , pdf-file
Lecture 8, Tuesday, April 24, 2012, Speaker: Ricardo De Sousa
Topic: p-adic numbers
Exercises from Lecture 8 ps-file , pdf-file
Lecture 9, Tuesday, May 8, 2012, Speaker: Amide Dervishi
Topic: Sums of squares.
Exercises from Lecture 9 ps-file , pdf-file
Lecture 10, Tuesday, May 15, 2012, Speakers: Kathleen Franzen and Ricardo De Sousa
Topic: The Tonelli-Shanks algorithm.
Exercises from Lecture 10 ps-file , pdf-file
Lecture 11, Tuesday, May 22, 2012, Speakers: Amide Dervishi and Mike Schomer
Topic: Discrete logarithm and the baby-step giant-step algorithm.
Exercises from Lecture 11 ps-file , pdf-file


Please let me know about misprints and/or mistakes found in the exercises and the lectures.

Some relevant links:


A Friendly Introduction to Number Theory.


Literature

Main books:
Silverman, Joseph H. - A friendly introduction to number theory, Upper Saddle River, N.J. : Pearson, 2006, 264 p.
Müller-Stach, Stefan; Piontkowski, Jens. - Elementare und algebraische Zahlentheorie. Ein moderner Zugang zu klassischen Themen, Wiesbaden: Vieweg. 2011, ix, 240 p.
Baldoni, Maria Welleda; Ciliberto, Ciro; Piacentini Cattaneo, Giulia Maria. - Elementary number theory, cryptography and codes, Berlin : Springer, 2009. - 522 p.
Cohen, Henri. - A course in computational algebraic number theory, Berlin: Springer, 1995, 534 p.
Cox, David A. - Primes of the form $x^2+ny^2$. Fermat, class field theory, and complex multiplication, Wiley-Interscience Series in Pure and Applied Mathematics. New York, NY: Wiley, 1997, xi, 351 p.
Additional books:
Neukirch, Jürgen. - Algebraic number theory; transl. from the German by Norbert Schappacher, Berlin: Springer, 1999, 571 p.
Samuel, Pierre. - Algebraic theory of numbers. (Translated from the French by Allan J. Silberger), Boston, Mass.: Houghton Mifflin Co., 1970, 109 p.
Stillwell, John. - Elements of number theory, New York, NY: Springer. 2003, xii, 254~p.
Pollard, Harry. - The theory of algebraic numbers, Carus Mathematical Monographs, No.9. New York: John Wiley & Sons. 1950, XII, 143 p.