Algebraic Number Theory
Math 514A Fall 2007
Problem Set 6
Due: Tuesday, Oct. 16th
1. Prove or disprove: Z(p)∼
=Zp, where Z(p)is the localization of Zat the prime ideal
(p), i.e. Zp={g
h|g, h ∈Z, p -h}. What about Qand Qp?
2. For a given prime number p, consider Σ = {0,1, ..., p −1}as a system of represen-
tatives for the elements of Fp. Find the representations of the form Piaipiwith
ai∈Σ of the following p-adic numbers:
a) −1∈Z3,2
5∈Z7;
b) The 4th roots of unity in Z5.
3. Show that the equation x2= 2 has a solution in Z7.
4. For each 0 < a < p, consider the sequence {xn}nin Qpwith xn=apn. Show the
following:
a) {xn}nis a Cauchy sequence in Qp.
b) Set ζa= lim xn. Then ζais a representative for ain Zp, and ζa∈µQp.
5. Prove or disprove: As abstract fields R∼
=Qp? What about Qpand Qqfor distinct
primes pand q?
6. Solve the following approximation problems:
a) K=Q,S={νpi|i= 1,2,3,p1= 2,p2= 3,p3= 5},x1=1
2, x2=1
3, x3= 1.
For every natural number nand n1=n2=n3=nfind solutions x∈Ksuch
that νpi(x−xi)≥nifor i= 1,2,3.
b) K=Q(t), S={νpi|i= 1,2, νp1=νt, νp2=νt−1},x1=t, x2= 1. For
every natural number nand n1=n2=nfind solutions x∈Ksuch that
νpi(x−xi)≥nifor i= 1,2.