Math 514 homework #21 Due date: 19951108 D. J. Bernstein 19951101 Let φ be a monic polynomial with integer coefficients. Factor φ over the complex numbers as ∏_{i}(x − α_{i}); here α_{1}, α_{2}, …, α_{n} are the roots of φ, counted by multiplicity. The discriminant of φ, written disc φ, is ∏_{i>j}(α_{i} − α_{j})^{2}. This is independent of the order of the roots. Exercise 63. Prove that disc φ = (−1)^{n(n−1)/2} result(φ, φ′), where n = deg φ. (Hence disc φ is an integer in φZ[x] + φ′Z[x].) Exercise 64. Prove that Φ_{m} is irreducible in Z[x]. (Hint: Let φ be the monic irreducible divisor of Φ_{m} such that φ(ζ_{m}) = 0. For which positive integers a is it true that φ(ζ_{m}^{a}) = 0?)