Math 514 homework #10 Due date: 19951004 D. J. Bernstein 19950927 The following exercise is a precise version of the statement “R mod I + J equals R mod J mod I.” Exercise 33. Let f be a surjective ring homomorphism from R to S. Let I be an ideal of R. Consider the surjective map g : R → S/f(I) defined by g(x) = f(x) mod f(I). Prove that the kernel of g is I + ker f. (Hence g induces an isomorphism between R/(I + ker f) and S/f(I).) Let A be a ring, and let J be an ideal of A. The norm of J from A to Z, written Norm_{A:Z} J or simply Norm J, is the size of A/J. Exercise 34. Let α be a complex number such that α^{3} = α + 1. Determine all the prime ideals of Z[α] with norm smaller than 27. In the following three exercises R is a Dedekind domain and P is a nonzero prime ideal of R. For any ideal I of R we define ord_{P} I = sup {n ∈ Z : I ⊆ P^{n}}. Exercise 35. Let I be an ideal. Prove that ord_{P} I = ∞ if and only if I is zero. Exercise 36. Let e be a function that assigns a nonnegative integer to each nonzero prime ideal Q of R. Assume that e(Q) = 0 for all but finitely many Q. Prove that ord_{P} ∏_{Q} Q^{e(Q)} = e(P). Exercise 37. Let I and J be ideals. Prove that ord_{P} IJ = ord_{P} I + ord_{P} J.