QUALIFYING EXAMINATION
AUGUST 2005
MATH 554  Dr. C. Wilkerson
There are eight problems, each worth 25 points for a total of 200 points.Unless otherwise stated,
show all necessary work. All rings are assumed to be commutative rings with a multiplicative iden
tity element.
I. (a) Let
A
be a ﬁnite abelian group of order 9
*
256. Let
φ
n
:
A
→
A
be the group homo
morphism that sends
x
→
nx
, for any integer
n
. The following information is known about ker(
φ
n
)
n
#ker(
φ
n
)
#ker(
φ
2
n
)
#ker(
φ
3
n
)
2
8
64
256
3
3
9
9
Deduce the structure of
A
as a direct sum of cyclic groups of prime power order. Give the in
variant factors for
A
.
(b) Let
V
be an 8 dimensional vector space over a ﬁeld
K
and let
ψ
∈
End
K
(
V
). Suppose that
the kernel of (
ψ

5)
j
has dimension
k
over
K
and that the following is known about
k
: for
j
=1
,
k
= 4; for
j
=2
,
k
= 7, and for
j
=3
,
k
= 8. Write down the rational canonical form and Jordan
canonical form for
ψ
.
II. (a) Deﬁne the concepts of Euclidean domain, PID, and UFD.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '09
 Math, Vector Space, Ring, Abelian group

Click to edit the document details