ASSIGNMENT 6  MATH235, FALL 2007
Submit by 16:00, Monday, October 29
1. Calculate the following:
(1)
(2
19808
+ 6)

1
+ 1 (mod 11).
(2)
12
,
12
2
,
12
4
,
12
8
,
12
16
,
12
25
all modulo 29. (Hint: think before computing).
2. Use the Euclidean algorithm to find the gcd of the following pairs of polynomials and express it as a
combination of the two polynomials.
(1)
x
4

x
3

x
2
+ 1 and
x
3

1 in
Q
[
x
].
(2)
x
5
+
x
4
+ 2
x
3

x
2

x

2 and
x
4
+ 2
x
3
+ 5
x
2
+ 4
x
+ 4 in
Q
[
x
].
(3)
x
4
+ 3
x
3
+ 2
x
+ 4 and
x
2

1 in
Z
/
5
Z
[
x
].
(4)
4
x
4
+ 2
x
3
+ 3
x
2
+ 4
x
+ 5 and 3
x
3
+ 5
x
2
+ 6
x
in
Z
/
7
Z
[
x
].
(5)
x
3

ix
2
+ 4
x

4
i
and
x
2
+ 1 in
C
[
x
].
(6)
x
4
+
x
+ 1 and
x
2
+
x
+ 1 in
Z
/
2
Z
[
x
].
3. Consider the polynomial
x
2
+
x
= 0 over
Z
/n
Z
.
(1)
Find an
n
such that the equation has at least 4 solutions.
(2)
Find an
n
such that the equation has at least 8 solutions.
4. Is the given polynomial irreducible:
(1)
x
2

3 in
Q
[
x
]? In
R
[
x
]?
(2)
x
2
+
x

2 in
F
3
[
x
]? In
F
7
[
x
]?
5. Find the rational roots of the polynomial 2
x
4
+ 4
x
3

5
x
2

5
x
+ 2.
6. Recall that for the ring
Z
a complete list of ideals is given by (0)
,
(1)
,
(2)
,
(3)
,
(4)
,
(5)
, . . .
, where (
n
)
is the principal ideal generated by
n
, namely, (
n
) =
{
na
:
a
∈
Z
}
. Find the complete list of ideals of the
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 Fall '07
 Goren
 Math, Algebra, Ring, Euclidean domain, Principal ideal domain

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