Math 514 homework #19 Due date: 19951103 D. J. Bernstein 19951027 Exercise 58. 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}B^{d} ∏_{i,j}(α_{i} − β_{j}). (Hint: Consider the roots as formal variables. In other words, map the multivariate polynomial ring Z[Y, Z, y_{1}, …, y_{d}, z_{1}, …, z_{e}] to R.) Exercise 59. Let l be a prime number not dividing m. Prove that x^{m} − 1 is squarefree in (Z/l)[x].