October Lectures

Week of October 2-6

Oct 2: Decoding algorithm for 2-error-correcting BCH codes. The multiplicative group of the nonzero elements in a finite
field is cyclic. Primitive elements. Representation of elements using primitive elements. Example of decoding procedure.
Oct 4: Review of rings, integral domains and characteristic of a ring. Finite fields have prime characteristic. Properties.
The ring of polynomials. Basic properties about the degree. [A2 handed out.]

Week of October 9-13

Oct 9: No lecture this day.
Oct 11: The ring of polynomials and relations with the ring of the coefficients. Division algorithm. Greatest common divisors
and Euclidean algorithm. Examples. Subrings and ideals. Examples.

Week of October 16-20

Oct 16: Ideals, principal ideals, maximal ideals, prime ideals and principal ideal domains. Characterizations of these structures.
Oct 18: More on characterizations of ideals. The ring of polynomials is a principal ideal domain.
A polynomial is irreducible if and only if the residue class ring is a field.

Week of October 23-27

Oct 23: Unique factorization in a ring of polynomials. Prime fields. Extension fields. Degree of the extension. [A3 handed out.]
Oct 25: Splitting field. Existence and uniqueness of splitting fields. Characterization of finite fields.
The number of elements of a finite field. Existence and uniqueness of finite fields.