Math 514 homework #11 Due date: 19951006 D. J. Bernstein 19950928 Let D be a domain. A function v : D → R ∪ {∞} is called a valuation on D if it satisfies the following properties: (1) v(x) = ∞ if and only if x = 0; (2) v(xy) = v(x) + v(y); (3) v(x + y) ≥ min {v(x), v(y)}. Exercise 38. Let D be a Dedekind domain. Let P be a nonzero prime ideal of D. Consider the function v : D → Z ∪ {∞} defined by v(x) = ord_{P} xD. Prove that v is a valuation on D. Exercise 39. Let D be a domain, let F be the field of fractions of D, and let v be a valuation on D. Prove that there is a unique valuation w on F that agrees with v on D. Exercise 40. What is the length of Z/20 over Z?