Fall 2009
MAT 322: ALGEBRA WITH GALOIS THEORY
PROBLEM SET #5
Week
#5: October 19th  October 25th.
Topics
: Rings, fields, ideals, quotient rings, Chinese remainder theorem, integral
domains, the field of fractions, PID’s, UFD’s, Euclidean domains, polynomials,
power series, Laurent series, Eisenstein’s criterion.
Read
: FGT [16] (also a good idea to check out Lang’s
Algebra
, Ch. II)
Problems
due Friday, October 23rd, at 4:00 pm in Martin Luu’s mailbox:
•
Problem 1
:
Let
F
4
be a field with four elements
{
0
,
1
, x, y
}
. Write down
the tables for addition and multiplication, hence showing
F
4
is unique.
•
Problem 2
:
Let
D
n
be the dihedral group of order
2
n
, where
n >
2
. Pick
generators
r
and
s
such that
r
n
=
e
,
s
2
=
e
, and
rs
=
sr

1
. Write
n
as
2
m
k
, where
k
is odd. For each
i
≤
m
, show that
Z
i
(
D
n
)
is generated by
r
n/
2
i
. Moreover, verify that
Z
i
(
D
n
) =
Z
m
(
D
n
)
for every
i
≥
m
.
•
Problem 3
:
Let
R
be an integral domain: A commutative ring such that
R
= 0
,
∀
a, b
∈
R
:
ab
= 0
⇒
a
= 0
∨
b
= 0
.
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 '09
 CLAUSSORENSEN
 Algebra, Remainder Theorem, Fractions, Remainder, Ring, Galois theory, Commutative ring, ideals I1 I2

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