Topics covered in the class
Sept. 5: Introduction, Fields.
Sept. 10: Characteristic, Binomial Theorem, isomorphism.
Sept. 12: Isomorphism, prime fields, polynmial rings.
Sept. 17: Polynomial rings, Division algorithm, gcd, Euclidean algorithm
Sept. 19: Irreducible polynomials, unique factorization, residue class rings
Sept. 24: Residue class fields, fields extensions.
Sept. 26: Fields extensions; linear codes.
Assign#1 is out.
Oct. 1: Linear codes.
Oct. 3: Syndrome and Hamming codes. Multiplicative group of a finite field
Oct. 8: review of cyclic groups; multiplicative group of finite fields; primitive elements
Oct. 10: Gauss algorithm, size of a finite field, Mobius functions,
Oct. 15: Mobius functions, existence of irreducible polynomials
Oct. 17: subfields, a distinction between finite fields with odd characteristic and even characteristic
Assign #2 is out.
Oct. 22: automorphisms, characteristic polynomials, minimal polynomials.
Oct. 24: midterm
Oct. 29: no class
Oct. 31: no class
Nov. 5: minimial polynomials, primitive polynomials
Nov. 7: period of polynomials, trace and norm.
Nov. 12 Trace and Norm.
Nov. 14. Bases; Berlekamp's algorithm
Nov. 19. Berlekamp's algorithm
Nov. 21. rationale , factorization of x^n -1.
Nov. 26. cyclic codes.
Nov. 28. Cyclic codes and Hamming codes.
Dec. 3. double error correcting BCH codes,
Dec. 5 BCH codes with designed distance, Reed-Solomon codes.
Assignment # 1
Assignment # 2