Math 515 homework #12 Due date: 19980304 D. J. Bernstein 19980225 Exercise 30. Let R be a finitely generated Z-algebra. Prove that R is finite if R is a field. (Hint: Z is a Jacobson ring.)

Newton’s method in the 10-adics. You may find Exercise 9 and Exercise 17 useful in the following exercise. Exercise 31. Let f ∈ Z[x] be a polynomial, and let f′ be its derivative. Let n and v be integers with f(n) ∈ 100Z and f′(n)v ∈ 1 + 10Z. Construct an integer m with f(m) ∈ 10000Z and m − n ∈ 100Z.