Math 514 homework #23 Due date: 19951115 D. J. Bernstein 19951107 Exercise 73. Let α be a root of some monic irreducible polynomial φ over Z. Let p be a prime number not dividing disc φ. Prove that p is unramified in Z[α]. Exercise 74. Let p be a prime number, and let k be a positive integer. Write q = p^{k}. Prove that Z[ζ_{q}] is a Dedekind domain. Exercise 75. Prove that Z[ζ_{m}] is a Dedekind domain for any positive integer m. (Hint: Induct on the number of distinct prime factors of m.)