Modern Algebra Fall 2006

Markus Schmidmeier
Mathematical Sciences
Florida Atlantic University

Modern Algebra
Welcome back to Algebra! This is Modern Algebra MAS 4301, we meet Mondays and Wednesdays 5:00 - 6:20 p.m. in SC 178.
Topics

This course introduces into the theory of numbers and of polynomials. Prime numbers are the building blocks of arbitrary numbers, and prime factorization provides the decomposition. Is a given number a prime number? We will use Fermat's Little Theorem as a test. The famous Fermat Last Theorem is a bit too hard for us to prove, but in this course you will have the opportunity to practise the writing of proofs. The key result in modular arithmetic is the Chinese remainder theorem. Modern Algebra is the prerequisite course for many exciting graduate and undergraduate courses in pure mathematics and in applications like Coding Theory and Cryptography. To give you the flavour, we will discuss the RSA public key cryptographic system.

Textbook

Lindsay N. Childs, A Concrete Introduction to Higher Algebra, 2nd edition, Springer Undergraduate Texts in Mathematics, ISBN 0-387-94484-2. I want to cover in particular the following sections.

Arithmetic in the integers 2A-D (induction); 2E-F (bases different from 10); 3A-C (the Euclidean algorithm); 4A-B (Fundamental Theorem of Arithmetic)

Modular arithmetic 5 (congruence and divisibility); 6 (Congruence classes, representatives)

Rings and fields 8A-C (axioms, integers modulo m, homomorphisms); 9A-C (Fermat's and Euler's theorem)

The RSA cryptosystem 10B

The Chinese Remainder Theorem 12A-B (CRT, products of rings)

Polynomials 14 (Ring of polynomials), 15A-D (Euclid's algorithm, irreducible polynomials)

Finite fields 28A-B (Arithmetic in F[x]/(p))

Credit

Quizzes: Every week there will be homework assignments. The quiz of about 20 minutes on Wednesday may be related to the homeworks. The nine best quizzes count for 40% of the grade.

Midterm Exam: The midterm exam on Monday, October 9, will count for 20% of your grade.

Final Exam: The final exam on December 4 (probably 5:00 - 7:30 p.m.) is comprehensive. It will count for 40% of your grade.

Contact Me

Office hours: MWF, 11-12 a.m. in SE 230, or after class.

Course Web Page: http://www.math.fau.edu/schmidme/algebra06.html

E-mail: markus@math.fau.edu.

Last modified: by Markus Schmidmeier

Arithmetic in the integers	2A-D (induction); 2E-F (bases different from 10); 3A-C (the Euclidean algorithm); 4A-B (Fundamental Theorem of Arithmetic)
Modular arithmetic	5 (congruence and divisibility); 6 (Congruence classes, representatives)
Rings and fields	8A-C (axioms, integers modulo m, homomorphisms); 9A-C (Fermat's and Euler's theorem)
The RSA cryptosystem	10B
The Chinese Remainder Theorem	12A-B (CRT, products of rings)
Polynomials	14 (Ring of polynomials), 15A-D (Euclid's algorithm, irreducible polynomials)
Finite fields	28A-B (Arithmetic in F[x]/(p))