Appendix F / F.1

F.1 AND STRUCTURE OF GALOIS FIELDS

A Galois field of order q (denoted GF(q)) contains exactly q elements. The order q must be a power of a prime, that is, q = p^m, where p is a prime number called the characteristic of the field, and m is a positive integer. The characteristic of GF(pᵐ) is p, meaning that adding the multiplicative identity to itself p times yields zero (that is, that arithmetic “wraps around” after p additions of 1); equivalently, p·1 = 0 within the field.

When m = 1, the field is called a prime field and consists simply of the integers {0, 1, 2, …, p − 1} with addition and multiplication defined modulo p. When m > 1, the field is called an extension field, and its elements are polynomials of degree less than m with coefficients in GF(p), defined modulo an irreducible polynomial (often chosen to be primitive) of degree m.