November Lectures
Week of October 30 - November 3
Nov 1: Subfield criterion. The multiplicative group of the
nonzero elements in a finite field is cyclic.
Nov 3: Primitive elements: Gauss' algorithm. Correctness and examples.
Week of November 6-10
Nov 6: The Moebius function and properties. The number of
irreducible polynomials of certain degree. Examples.
Nov 8: Fast algorithms for testing the irreducibility of a
polynomial. Examples. [A4 handed out.]
Week of November 13-17
Nov 13: Factorization of polynomials: applications;
complete factorization, and squarefree factorization.
Nov 15: Distinct-degree factorization, equal-degree
factorization (Cantor and Zassenhaus) and examples.
Week of November 20-24
Nov 20: Cyclic codes: definition, examples and characterization
as ideals.
Nov 22: Cyclic codes (cont.): generator polynomial and matrix.
Minimal polynomials: definition and properties. [A5 handed out.]
Week of November 27 - December 1
Nov 27: More properties of minimal polynomials. Cyclotomic
cosets; examples. Check polynomial and parity check matrix.
Nov 29: Cyclotomic cosets and factors of x^n - 1. Hamming
codes as examples of cyclic codes. Coments about the exam.
Week of December 4
Dec 4: 2-error-correcting BCH codes as example of cyclic codes.
Computing minimal polynomials. Course overview.
Dec 7: Final exam.