Math 793

Summer 2001

Homework 3

 

Turn in at least two by Monday, July 2, 2001.

 

  1. Consider the ring  where  is a square-free integer. Consider the norm map given by  Prove the following properties of the norm:

 

a)

b)  is a unit if and only if

c)  if and only if

 

  1. Consider the ring  where  For this problem, you may assume that  is a UFD.

a)      Find all of the units in  and

b)      Show that for any element  there is a unit in  such that

c)      Show that  is atomic (hint: norm) and that any irreducible in  is a prime element in

d)      Use the previous to show that any two factorizations of a nonzero nonunit in  have the same length (such a ring is called is called a half-factorial domain or HFD).

e)      Show that  is not a UFD.

 

  1. Let be a field.

a)      Show that  is a Euclidean domain.

b)      Classify the following rings as PIDs or UFDs or neither: