Math 4108 FInal Exam, Spring 2010
May 5, 2010
1. Define the following terms
a. Euclidean Domain.
b. Constructible number.
c. Splitting field.
d. JordanHolder Theorem.
e. Associative algebra.
2. Show that in a principal ideal domain, any nested chain of ideals
I
1
⊆
I
2
⊆
I
3
· · ·
must necessarily satisfy
I
k
=
I
k
+1
=
· · ·
,
for some
k
≥
1.
3. Suppose that
K
and
L
are finite extensions of a field
F
. Let
M
be a
finite extension of
F
such that
K, L
⊆
M
, and such that
[
M
:
F
]
<
[
K
:
F
]
·
[
L
:
F
]
.
Prove that
F
is properly contained in
K
∩
L
.
4. Let
K
be the splitting field over
Q
of the algebraic number
α
=
radicalBig
2 +
√
3
.
Compute the Galois group of
K/
Q
– i.e. describe Gal(
K/
Q
).
1
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5. Suppose that
f
(
x
) is a particular polynomial of degree 5 whose Galois
group (over
Q
) has order exceeding 30. Prove that
f
(
x
) is not solvable
in the radicals.
6. Determine the number of degree 11 monic polynomials in
F
3
[
x
] that
irreducible (write down the exact number).
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 Spring '10
 Staff
 Algebra, Group Theory, splitting field

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