Midterm 1, Math 4108, Spring 2010
March 1, 2010
1.
Define the following terms.
a. Algebraic extension.
b. Unique Factorization Domain.
c. Splitting field.
d. Eisenstein’s criterion.
e. algebraic number.
2.
Compute the degree of the extension
F
[
x
]
/
(
x
2

2) over
Q
, where
F
=
Q
(
i
), where
i
2
=

1. Explain your answer.
3.
Determine the gcd of the polynomials
f
(
x
) =
x
5
+ 3
x
3
+ 2
x
2
+ 2
x
+ 2 and
g
(
x
) =
x
5
+
x
4
+ 3
x
3
+ 4
x
2
+ 4
x
+ 2, where we think of
f, g
∈
F
7
[
x
].
4.
Suppose that
K
is an algebraic extension of a characteristic 0 field
F
, in
which every
α
∈
K
satisfies some polynomial of degree at most
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 Spring '10
 Staff
 Algebra, Eisenstein, primitive element theorem, Unique Factorization Domain.

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