MTHSC 851/852 (Abstract Algebra)
Dr. Matthew Macauley
HW 12
Due Monday, September 7, 2009
(1) (a) Let
R
be a UFD (unique factorization domain, commutative), and let
d
a nonzero
element in
R
. Prove that there are only ﬁnitely many principal ideals in
R
that
contain
d
.
(b) Give an example of a UFD
R
and a nonzero element
d
∈
R
such that there are in
ﬁnitely many ideals in
R
containing
d
. [No proof is required for this part; however,
you must describe not only
R
and
d
, but also an inﬁnite family of ideals containing
d
.]
(2) Suppose
f
(
x
) = 1 +
x
+
x
2
+
···
+
x
p

1
, where
p
∈
Z
is prime.
(a) Show that
f
is irreducible in
Q
[
x
]. [Hint: Write
f
(
x
) = (
x
p

1)
/
(
x

1), and substitute
x
+ 1 for
x
].
(b) Show that
(
p
k
)
=
∑
k
+1
i
=1
(
p

i
p

k

1
)
for all
k < p
.
(3) (a) All of the following rings
R
i
, for
i
= 1
, . . . ,
6 are additionally
C
vector spaces. In each
case, compute the vector space dimension by explicitly giving a basis for
R
i
over
C
in each case.
R
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 Fall '08
 Staff
 Algebra, Euclidean algorithm, Ri, Euclidean domain, Principal ideal domain, Matthew Macauley HW

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