advertisement

Math 426, Winter 2014, Term 1 Problems on p-adic numbers and the Cantor set Problem 1. Consider Q with the p-adic valuation defined by |x|p = p− ordp (x) , where ordp (x) = α ∈ Z if u x = pα , v where u, v ∈ Z, p - u, p - v. Note that the value group, i.e., the set of valuations of the non-zero numbers is pZ . A Cauchy sequence is a sequence (xi ) of rational numbers such that for every > 0, there exists an n, such that for all i, j ≥ n: |xi − xj |p < . The Cauchy sequences form a ring by componentwise addition and multiplication. Define a valuation on the Cauchy sequences by |(xi )|p = lim |xn |p . n→∞ Prove that this is well-defined, and satisfies the two properties 1. |(xi )(yi )|p = |(xi )|p |(yi )|p , 2. |(xi ) + (yi )|p ≤ max(|(xi )|p , |(yi )|p ) . Deduce that the set of Cauchy sequences whose valutation is 0 is an ideal, and that the quotient of the Cauchy sequences by this ideal is a field. This field is Qp , the field of p-adic numbers. Deduce also, that | · |p passes to Qp , and that for Qp we have 1. |x|p = 0 if and only if x = 0, 2. |xy|p = |x|p |y|p 3. |x + y|p ≤ max(|x|p , |y|p ) . 1 Problem 2. Note that every series ∞ X an p n , n=N where an ∈ {0, 1, . . . , p − 1}, for all n, N ∈ Z, and aN 6= 0, is a Cauchy seqence of partial sums of rational numbers and therefore defines a p-adic number. Prove that every p-adic number can be written as a limit of such a series, in a unique way. Write the rational number 13 this way, for p = 2. Problem 3. Do Exercise 6 of §27 in Munkres on the Cantor set. Problem 4. Prove that the Cantor set is homeomorphic to the 2-adic integers Z2 = {x ∈ Q2 : |x|p ≤ 1} . (Note that Zp is the closure of Z in Qp .) 2