Math 515 homework #18 Due date: 19980424 D. J. Bernstein 19980417 In the following exercise, R is a Noetherian ring, P is a prime ideal of R, and n is the height of P. Recall that P is nonsingular if the ideal P R_{P} of the localization R_{P} is generated by n elements. Exercise 50. Prove that P is nonsingular if and only if len_{R}(P/P^{2}) = n. A Noetherian domain of dimension 1 is normal if and only if all of its maximal ideals are nonsingular. A normal Noetherian domain of dimension 1 is called a Dedekind domain.

Dedekind’s criterion. In the following exercises, φ ∈ Z[x] is a monic irreducible polynomial; A is the quotient ring Z[x]/φ; p ∈ Z is a prime number; h ∈ Z[x] is a monic polynomial such that h mod p is an irreducible factor of φ mod p in (Z/p)[x]; and d is the degree of h. Write I = pZ[x] + hZ[x] and Q = pA + hA. Then Z[x]/I is isomorphic to the field (Z/p)[x]/h, so I is a maximal ideal of Z[x] containing φ, so Q is a maximal ideal of A. The following exercises show how to test whether len_{A}(Q/Q^{2}) = 1, i.e., whether Q is nonsingular. Exercise 51. Prove that len_{Z[x]}(I/I^{2}) − len_{Z[x]}(Q/Q^{2}) is 0 if φ ∈ I^{2}, 1 if φ ∉ I^{2}. Exercise 52. Define M = Z+Zx+⋯+Zx^{d−1}. Prove that I/I^{2} is isomorphic to (pM/p^{2}M)⊕(hM/phM) as a Z-module. (Hint: Prove that Z[x] = M ⊕hZ[x]; thus hZ[x] = hM ⊕h^{2}Z[x] and Z[x] = M ⊕hM ⊕h^{2}Z[x].) Exercise 53. Let k be a finite field. Let V be a k-module. Prove that len_{Z} V = (len_{Z} k) len_{k} V. Exercise 54. Prove that len_{Z[x]}(I/I^{2}) = 2. Exercise 55. Prove that len_{A}(Q/Q^{2}) > 1 if and only if (h mod p)^{2} divides φ mod p and h mod p^{2} divides φ mod p^{2}. For example, put A = Z[x]/(x^{2} − 5). The maximal ideal 2A + (x − 1)A is singular: (x − 1)^{2} divides x^{2} − 5 over Z/2 and x − 1 divides x^{2} − 5 over Z/4. Thus A is not a Dedekind domain.