Michel Waldschmidt

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.

  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

    Dummit, D. S. and Foote, R. M. - Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1998. ยง14.3: 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).