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From April 7 to 17, 2009,
Ho Chi Minh University of Science HCMUS

#### Finite fields: some applications

*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

How to use finite fields for problems concerning infinite fields, following Jean-Pierre Serre.

*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.

Jean-Pierre Serre. - How to use finite fields for problems concerning infinite fields; arXiv:0903.0517v2 [math.AG].

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

First course, April 8, 2009, pages 1-12 (updated April 16, 2009)Second course, April 10, 2009, pages 13-18 (updated April 16, 2009)

Third course, April 13, 2009, pages 19-26 (updated April 18, 2009)

Fourth course, April 15, 2009, pages 27-40 (updated April 18, 2009)

Fifth course, April 18, 2009, pages 41-45 (updated April 18, 2009)

Playing with cards and hats - an introduction to error correcting codes. Powerpoint presentation on April 18, 2009 (updated April 18, 2009)

Exercises with corrections (updated April 26, 2009)