Math 514 homework #20 Due date: 19951106 D. J. Bernstein 19951030 Exercise 60. Let A, B, α_{1}, …, α_{d}, β_{1}, …, β_{e} be elements of some ring R. Write f = A ∏_{i}(x − α_{i}) and g = B ∏_{j}(x − β_{j}). Prove that result(f, g) = A^{e} ∏_{i} g(α_{i}). Exercise 61. How is result(f, g) related to result(g, f)? Exercise 62. Let l be a prime number not dividing m. Let ζ be an mth root of unity. Let φ, h be monic irreducible polynomials over Z such that φ(ζ) = 0 and h(ζ^{l}) = 0. Prove that φ = h. (Hint: Define H(x) = h(x^{l}); note that H mod l = (h mod l)^{l} in (Z/l)[x].)