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Solutions
Exam 2
Instructions.
Answer each of the questions on your own paper, and be sure to show your work
so that partial credit can be adequately assessed. Put your name on each page of your paper.
1. [
26
Points]
(a) Verify that
p
(
x
) =
x
2
+1 is irreducible over the ﬂeld
Z
3
, but it is reducible over the ﬂeld
Z
5
.
I
Solution.
Over
Z
3
,
p
(0) = 1,
p
(1) = 2, and
p
(2) = 2. Thus,
p
(
x
) has no roots in
Z
3
,
and since deg
p
(
x
) = 2 it follows that
p
(
x
) is irreducible.
Over
Z
5
,
p
(2) = 2
2
+1 = 0 so
x
2
+1 = (
x
¡
2)(
x
¡
3)
2
Z
5
[
x
] and hence is reducible.
J
(b) If
F
=
Z
3
[
x
]
=
(
x
2
+ 1) then
F
can be represented by
F
=
f
a
+
bt
:
a; b
2
Z
3
g
;
where
t
is the congruence class of
x
.
F
is a ﬂeld since
x
2
+ 1 is irreducible over
Z
3
.
i. How many elements are there in the ﬂeld
F
. List all of them.
I
Solution.
j
F
j
= 3
2
= 9.
F
=
f
0
;
1
;
2
; t; t
+ 1
; t
+ 2
;
2
t;
2
t
+ 1
;
2
t
+ 2
g
.
J
ii. Let
z
=
t
+ 1
2
F
. Compute
z
2
,
z
4
, and 1
=z
in the ﬂeld
F
. Express each of your
answers in the standard form
a
+
bt
for some
a; b
2
Z
3
.
I
Solution.
z
2
= (
t
+ 1)
2
=
t
2
+ 2
t
+ 1 =
¡
1 + 2
t
+ 1 = 2
t
,
z
4
= (
z
2
)
2
= (2
t
)
2
=
t
2
=
¡
1 = 2, and
1
z
=
1
t
+ 1
=
1
t
+ 1
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This document was uploaded on 12/28/2011.
 Fall '09

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