ABSTRACT
In this chapter, we describe factorization of u n − 1 in terms of its cyclotomic factors over the finite fields GF(p) and GF(q), q = p m . GF(p) is the field with elements {0, 1, ...,p –1}, where the arithmetic operations defined mod p. GF(q) is obtained from GF(p) by defining as elements all polynomials of degree up to m − 1 with coefficients in GF(p). The arithmetic operations of ‘+’ and ‘−’ are defined mod P(v), P(v) being an irreducible polynomial over GF(p), deg(P(v)) = m. Thus, GF(q) is a polynomial extension of GF(p). Note that we have changed the indeterminate from u to v in order to distinguish the polynomial representation of the elements of GF(q) from the remainder of our discussion on polynomial algebra. GF(q) exists for all values of prime p and m. In other words, irreducible polynomials of degree m exist over GF(p) for all prime p and m > 0. In fact, for a finite field having a elements, a must have the form a = p m .
