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Outline of an eight-chapter book titled Galois Fields, in the Galois Book folder of original material from Philips Electronics. It lists topics from groups, rings and ideals through polynomials, construction of GF(q=p^m), primitive and minimum polynomials, and field tables. Later chapters cover cyclic codes (BCH, Hamming, Reed-Solomon, CRC) and matrix representations of Galois fields.
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Galois Fields
Chapter 1: A Review of Modern Algebra
groups, subgroups, cosets, normal subgroups, factor group, cyclic subgroups,
order, generators, additive cyclic subgroups,
rings, ideals, residue classes, residue class rings, principle ideals,
the example Zn = modulo-n, the Division Algorithm,
the Factorization Theorem, the Euclidean Division Algorithm, relative primeness,
the fact that Z/(n) = Zn is a field if n is prime
Chapter 2: The Galois Fields GF(p)
GF(p) = Zp , facts about
Chapter 3: Polynomials
the field F underlying a polynomial, polynomials form a ring,
quotient and remainder polynomials, irreducible polynomials,
the construction GF(pm ) = Poly[ x,GF(p)] / (f(x),m),
ground field and extension field, f(x) must be irreducible.
Chapter 4: The Galois Fields GF(q=pm)
m-tuple notation, facts about the order of an element, roots of f(x),
{ GF(q) - 0, • } is cyclic, exponents, primitive elements a,
all GF(q) field elements are roots of xq-1- 1, examples
Chapter 5: Minimum Polynomials
minimum polynomials, monic, primitive polynomials,
period, conjugate set of a, finding primitive polynomials,
Chapter 6: The Galois Field Table
how to construct a complete GF(q) field algebra from any primitive polynomial
Chapter 7: Galois-Based Cyclic Codes
linear block codes, symbols, parity check, data words, code words, matrices G and H,
dual code, minimum distance d, error-correction of t-symbol errors in a codeword,
the syndrome, major codes, code history
cyclic codes, cyclic vs systematic basis, implementation of encoders, CRC,
proof that cyclic codes are cyclic, the standard array as an ideal,
the connection between cyclic codes and Galois Fields,
general BCH codes, narrow-sense BCH codes, Hamming codes,
Reed Solomon Codes, digital filters
Chapter 8: Matrix representations of Galois Fields
a review of matrix algebra, characteristic polynomial of a matrix
polynomials of a matrix, the Cayley-Hamilton theorem, ways to think of a remainder,
matrix representation of a Galois Field, the compansion matrix forms, examples