Math 515 homework #13 Due date: 19980309 D. J. Bernstein 19980302 Exercise 32. Eisenbud 7.1. Exercise 33. Eisenbud 7.13. (You may assume the axiom of choice. For the counterexample, take any sufficiently large ring, and consider any countably generated ideal that is not finitely generated.)

From Newton’s method to Hensel’s lemma. In the following exercises, R is a ring, P is an ideal of R, R̂ is the P -adic completion of R, and P̂_{n} is the kernel of the natural map R̂ → R/P^{n}. Note that P̂_{n}^{2} ⊆ P̂_{2n}. Exercise 34. Let f ∈ R[x] be a polynomial, and let f′ be its derivative. Let α be an element of R̂ with 1 ∈ f′(α)R̂ and f(α) ∈ P̂_{n} for some n ≥ 1. Construct an element β of R̂ with 1 ∈ f′(β)R̂, f(β) ∈ P̂_{2n}, and β − α ∈ P̂_{n}. Exercise 35. Let f ∈ R[x] be a polynomial, and let f′ be its derivative. Let α be an element of R̂ with 1 ∈ f′(α)R̂ and f(α) ∈ P̂_{n} for some n ≥ 1. Prove that there is a unique element β of R̂ with 1 ∈ f′(β)R̂, f(β) = 0, and β − α ∈ P̂_{n}.