Resultants of cyclotomic polynomials

A formula for Resx(ca(xb), (D(xd)) is given where Da(x) denotes the ath cyclotomic polynomial. This extends a result of Lehmer, Diederichsen, and Apostol. The nth cyclotomic polynomial, n > 1, is defined by On (x) = fj(x e 2nik/n) where the product is over all values of k relatively prime to n such that 1 I k < n. The polynomial On(x) is irreducible in Z[x] with degree (0(n) where (0 denotes the Euler phi-function. Lehmer [6], Diederichsen [4] and Apostol [ 1 ] have given formulas for Resx ('$m (x), ,n (x)) . Later Apostol [2] extended it to a formula for Resx (t m(ax), On(bx)) with the hope to shorten the 255page proof of the celebrated Feit-Thompson Theorem [5]. In this paper we derive a formula for Resx((Da(xb), Dc(xd)) extending the result of Lehmer, Diederichsen, and Apostol. In doing so, we first factorize (Da(xb) in Z[x] followed by an application of the chain rule for resultants. Throughout this paper, (a, b) will denote the greatest common divisor, and [a, b] the least common multiple, of positive integers a and b. We also use the notation alb to indicate that a divides b.