Math 514 homework #18 Due date: 19951101 D. J. Bernstein 19951024 Exercise 55. Let A, B, α_{1}, …, α_{d}, β_{1}, …, β_{e} be elements of some domain 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}) if all α_{i}’s and β_{j}’s are distinct. (Hint: Multiply Sylvester’s matrix by an appropriate Vandermonde-ish matrix.) The mth cyclotomic polynomial Φ_{m} ∈ C[x] is ∏_{a∈{0,1,...,m−1}, gcd{a,m}=1}(x − ζ_{m}^{a}). In other words, it is the product of x − ζ where ζ runs through all primitive mth roots of unity. Exercise 56. Prove that x^{m} − 1 = ∏_{d} Φ_{d} where d runs through the positive integers dividing m. Exercise 57. Prove that Φ_{m} ∈ Z[x].