Math 514 homework #22 Due date: 19951110 D. J. Bernstein 19951101 Exercise 65. Let p be a prime number, and let k be a positive integer. Write q = p^{k}. Prove that Φ_{q} = (x^{q} − 1)/(x^{q/p} − 1). Exercise 66. Let p be a prime number, and let k be a positive integer. Write q = p^{k}. Let R be a Dedekind domain inside some number field F. Let Q be a nonzero prime ideal of R[ζ_{q}]. Assume that Q is noninvertible. Prove that Q contains p. Exercise 67. Let α be a root of some monic irreducible polynomial φ over Z. Prove that disc φ = disc_{Q(α):Q} Z[α]. In the following five exercises, φ is a monic irreducible polynomial over Z and α is a root of φ. Calculate disc_{Q(α):Q} Z[α]. Exercise 68. φ = x^{2} + 1. Exercise 69. φ = x^{2} + 19. Exercise 70. φ = x^{3} − x − 1. Exercise 71. φ = x^{3} + x^{2} + 5x − 16. Exercise 72. φ = (x^{p} − 1)/(x − 1), where p is an odd prime number.