Math 420/620
Fall 1999
Final Exam
- Let
be a UFD, show that
is irreducible if and only if it is prime.
- Consider the ring Z
. We define the norm map
by ![](Image648.gif)
. Show the norm has the following properties:
for all ![](Image650.gif)
is a unit in ![](Image652.gif)
![](Image653.gif)
Use these facts to show that the ring Z
is not a UFD.
- Show the following conditions are equivalent:
is a PID
is a UFD and every nonzero prime ideal of ![](Image657.gif)
is maximal.
- a) Construct an example of a ring that has precisely
maximal ideals.
b) (G) If your example from a) is not a domain, then make a new example which is a domain (reverse the instructions if your example from a) is a domain).
- Consider that the polynomial ring
![](Image659.gif)
. Show that the following conditions are equivalent.
is a PID
is a Euclidean domain
![](Image662.gif)
is a field.
- (G)
Let
be a domain with quotient field
We say that
is integral if
is a root of a monic polynomial with coefficients in
Show that if
is a UFD and
is integral, then
- Let
be commutative with identity. Let
be any ideal and let
. Show that
is an ideal of
and that ![](Image676.gif)