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College of the Holy Cross, Fall 2009
Math 351 – Solutions to the practice problems for exam 3
#1.
Suppose that
a
is algebraic over
Q
and
f
(
x
) is a polynomial with coeﬃcients in
Q
. Show that
Q
(
f
(
a
)) is an algebraic extension of
Q
.
Since
f
(
x
) has coeﬃcients in
Q
, we have
f
(
x
) =
b
n
x
n
+
···
+
b
1
x
+
b
0
, where
a
i
∈
Q
. Thus
f
(
a
) =
b
n
a
n
+
···
+
b
1
a
+
b
0
, which is in
Q
(
a
) since
Q
(
a
) is closed under ﬁeld operations. Thus
Q
(
f
(
a
))
⊆
Q
(
a
). Since
Q
(
a
) is algebraic, it is ﬁnite (its degree over
Q
is equal to the degree of the
minimal polynomial of
a
over
Q
). Because
Q
(
f
(
a
))
⊆
Q
(
a
), we have [
Q
(
f
(
a
)) :
Q
]
≤
[
Q
(
a
) :
Q
], and
thus
Q
(
f
(
a
)) is ﬁnite. Since ﬁnite extensions are algebraic (Theorem 21.4), we have that
Q
(
f
(
a
))
is algebraic.
#2.
Let
E
=
Q
(
p
(3)
,
p
(5)), and assume that [
E
:
Q
] = 4. Find the elements of Gal(
E/
Q
) (recall

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