On the Cyclotomic Polynomial Φpq (X)

The mth cyclotomic polynomial 4>m(X) is defined to be fl(X;), where; ranges over the primitive mth roots of unity in C:. It is well-known that 4>m(X) is an irreducible polynomial in @[X] with degree q(m), where So denotes Euler's totient function. In particular, 4>m(X) is the minimal polynomial of; over ¢p. If mO denotes the largest square-free factor of m, then 4>m(X) = 4>mO(Xm/m0). Therefore, the computation of 4>m(X) can be reduced to the case when m = pq *me, where p, q, . . . are distinct primes. Of course, 4>p(X) = XP-1 + * +X + 1, so the next interesting case is when m = pq. Here are two explicit examples: