- 1, then Se is a coset of the subgroup S1 in the group of integers relatively prime to qd - 1 under multiplication modulo qd - 1, and represents the roots of another primitive polynomial. If e is not rel- atively prime to qd - 1, then Se is not the result of a formal coset construc- tion; however, all the distinct sets S are called cyclotomic cosets. Note that the d symbol, base q expansion of e and eqi modulo qd - 1 are related for all i by a cyclic shift, making the determination of the sets S a simple mat- ter in a base q number system. Example 5A.15. In GF(16), the root logarithm sets are given by S1, S3, S5, and S7, with S1 being the basic subgroup, S7 representing the one formal coset of S1, and S3 and S5 representing the remaining cyclotomic cosets. Using the element C of Table 5A.2 as the primitive element, the q-conjugate algorithm can be used to construct the polynomials corresponding to the above sets: So = {0}->z+1 Appendix 5A. Finite Field Arithmetic The following relation reduces the amount of work necessary to evaluate the irreducible polynomials associated with the root sets. Let (5A.61) be an irreducible polynomial whose roots have exponent e. Then the corre- sponding reciprocal polynomial, defined by (5A.62) is easily determined by reversing the coefficient order (and scaling if nec- essary). The roots of B(z) are the reciprocals of the roots of A(z) and have the same order. Hence, in Example 5A.15, knowing that C and C14 are reci- procals, and that mc(z) is Z4 Z + 1 is enough to conclude directly that 5A.9. THE TRACE FUNCTION The trace polynomial from GF(q") to GF(q) is defined as (5A.63) where the superscript and subscript on Tr have been added to clarify the fields under consideration. The trace in GF(q) of an element a in GF(q") is then defined as Trg"(a), i.e., the trace polynomial evaluated at a. For a given element a which is in several fields, Tra(a) will vary as the choice of q and/or qn varies. We shall drop the superscripts and subscripts on Trg(a) only when n and q are obvious in th