Department of Mathematical Sciences
Instructor: Daiva Pucinskaite
Algebraic Number Theory
September 26, 2016
Name:
Homework Assignment 5
1. Let p be a prime number. Consider the ring
nm
o
Z(p) :=
∈ Q | n is not divisible by p
n
m
1. Show:
∈ Z(p) is a unit if and only if neither m nor n are divisible by p.
n
2. Show: x ∈ Z(p) is a prime element if and only if there exists a unit u ∈ Z(p) such that
x = u · p.
3. Show:
If x ∈ Z(p) then there exists a unit u ∈ Z(p) , and m ∈ N such that x = u · pm .
4. Show:
The ideal q = pZ(p) of Z(p) generated by p is the (unique) maximal ideal of Z(p) .
5. Show: If I is an ideal of Z(p) then I = qm for some m ∈ N, and qm ⊆ qs if and only if
s ≤ m.
6. Show:
The map f : Z/pm Z → Z(p) /qm , with f (a + pm Z) = a + qm is an isomorphism.
2. Let A be a ring. Let a and b be relatively prime ideals in A. Show that for any m, n ∈ N
the ideals am and bn are relatively prime.
3. Let A be a Dedekind domain. Let a be a fractional ideal of A with da ⊆ A.
1. Show: da is an ideal of A.
2. Show: The map f : a → da, with f (a) = da is an A-module isomorphism.
3. Show: a is finitely generated.
( n
)
X
4. Show: The set
ai OL 6= 0 | n ∈ N, a1 , . . . an ∈ L is the set of fractional ideal of OL .
i=1
4. Show: The set S :=
ideals of Z.
1
Zm | m ∈ N, n ∈ N, gcd(m, n) = 1
n
is the set of fractional