Appendix F / F

F GALOIS FIELDS

In digital communications and coding theory, finite fields—also known as Galois Fields—play a foundational role. Named after the French mathematician Évariste Galois (1811–1832), these algebraic systems provide the mathematical framework for the construction and manipulation of error-control codes, such as CRC, Hamming codes, BCH codes, and Reed–Solomon codes. A Galois Field is a field that contains a finite number of elements, and it satisfies all the familiar arithmetic properties of the real numbers: closure, associativity, commutativity, distributivity, and the existence of additive and multiplicative identities and inverses. What makes Galois fields distinctive is that arithmetic is performed modulo a prime number or a polynomial, so all results remain within a finite set of possible values.