Abdus Salam School of Mathematical Sciences (ASSMS), Lahore (Pakistan)

A Course on Finite Fields.

From October 3 to 26, 2011

Abstract: The theory of finite fields is a beautiful and rich theory of mathematics, which does not require too much knowledge to start with, but which is deep and has a lot of applications, in particular to data transmission, cryptography and error correcting codes.

Syllabus:
  Cyclotomic Polynomials
       Cyclotomic Polynomials over the ring of integers
       Cyclotomic Polynomials over any ring
       Cyclotomic Polynomials over a finite field
       Proof of the irreducibility of the cyclotomic polynomials over the integers
  Error correcting codes
       Cyclic codes
       Hamming codes
       Generator matrix and check matrix

References:
    Dummit, D. S. and Foote, R. M. - Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1998. §14.3: Finite Fields, pp. 499-505.
    Lang, S. - Algebra, 3rd Ed.
    Lidl, R. and Niederreiter, H. - Introduction to Finite Fields and their Applications. Cambridge University Press; 2 edition (August 26, 1994)
    G.L. Mullen, C. Mummert. - Finite Fields and Applications, Student mathematical library, 41, AMS 2007.
    Zhe-Xian Wan. - Lectures on finite fields and Galois rings, Word Scientific Publishing Co. Pte. Ltd. 2003.
On the internet:
    William Chen- Discrete Mathematics, 201 pp. (web edition, 2008).
    Shoup, V. - A Computational Introduction to Number Theory and Algebra, Cambridge 2005. Second print editon, Fall 2008 (pdf file 3,5 Mo).

Notes of the course:     Finite Fields (pdf: 53 pages, updated 03/10/2011).