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College of the Holy Cross, Spring 2010
Math 352, Midterm 2
Monday, April 19
You may use any results from class or from the text, except for homework
exercises unless explicitly stated otherwise.
To get full credit for your answers, you must fully explain your reasoning.
1. Give an example of a degree 4 polynomial in
Z
5
[
x
] that has no multiple roots in any
extension of
Z
5
. Give an example of a degree 4 polynomial in
Z
5
[
x
] that has no roots
in
Z
5
but has multiple roots in some extension of
Z
5
.
Consider
f
(
x
) = 5
x
4
+
x
. We have
f
0
(
x
) = 1, and so clearly
f
(
x
) and
f
0
(
x
) can have
no common factor of positive degree (since
f
0
(
x
) has no factor of positive degree at
all). Thus by Theorem 20.5,
f
(
x
) has no multiple roots in any extension of
Z
5
. Note
that there is no requirement in Theorem 20.5 that the polynomial be irreducible.
Now consider
f
(
x
) = (
x
2
+ 2)
2
. By checking all the elements of
Z
5
, one sees that
f
(
x
) has no roots in
Z
5
. However,
f
0
(
x
) = 4
x
(
x
2
+ 2), and so
f
(
x
) and
f
0
(
x
) have
the common factor of
x
2
+ 2. Thus by Theorem 20.5,
f
(
x
) has multiple roots in some
extension of
Z
5
.
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View Full Document2. Let
α
be a root of the polynomial
f
(
x
) = 2
x
4
+ 6
x

3
∈
Q
[
x
].
(a) Find a basis for
Q
(
α
) as a vector space over
Q
(b) Write
α

1
as a linear combination of elements of your basis from part (a).
(c) Consider the set
S
=
{
c
1
α
+
c
0

c
0
,c
1
∈
Q
}
. Prove or disprove that
S
is a subspace
of
Q
(
α
).
(a) First note that by the Eisenstein criterion,
f
(
x
) is irreducible over
Q
. By Theorem
20.3, a basis for
Q
(
α
) as a vector space over
Q
is
{
1
,α,α
2
,α
3
}
, since the degree of
f
(
x
)
is 4.
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