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7. FIELD EXTENSIONS – FIRST LOOK
7.3.8.
Show that the splitting field
E
of
x
3
2
has dimension 6 over
Q
.
Refer to Exercise
7.2.8
.
7.3.9.
If
˛
is a fifth root of
2
and
ˇ
is a seventh root of 3, what is the
dimension of
Q
.˛; ˇ/
over
Q
?
7.3.10.
Find dim
Q
Q
.˛; ˇ/
, where
(a)
˛
3
D
2
and
ˇ
2
D
2
(b)
˛
3
D
2
and
ˇ
2
D
3
7.3.11.
Show that
f .x/
D
x
4
C
x
3
C
x
2
C
1
is irreducible in
Q
OExŁ
. For
p.x/
2
QOExŁ
, let
p.x/
denote the image of
p.x/
in the field
QOExŁ=.f .x//
.
Compute the inverse of
.
x
2
C
1/
.
7.3.12.
Show that
f .x/
D
x
3
C
6x
2
12x
C
3
is irreducible over
Q
. Let
be a real root of
f .x/
, which exists due to the intermediate value theorem.
Q
. /
consists of elements of the form
a
0
C
a
1
C
a
2
2
. Explain how to
compute the product in this field and find the product
.7
C
2
C
2
/.1
C
2
/
.
Find the inverse of
.7
C
2
C
2
/
.
7.3.13.
Show that
f .x/
D
x
5
C
4x
2
2x
C
2
is irreducible over
Q
. Let
be a real root of
f .x/
, which exists due to the intermediate value theorem.
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 Fall '08
 EVERAGE
 Algebra, Intermediate Value Theorem, KŒx

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