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 ) .
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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 .)
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