Ma
5
c
HOMEWORK
2
SOLUTION
SPRING 09
The exercises are taken from the text,
Abstract Algebra
(third edi
tion) by Dummit and Foote.
Page 545,
5
.
Suppose
K
is a splitting field over
F
for a collection
of
{
f
i
(
x
)
}
.
Let
f
be an irreducible polynomial over
F
with a root
α
∈
K
.
Let
β
be any other root of
f
.
By Theorem 8, there is an
isomorphism
σ
:
F
(
α
)
→
F
(
β
). Now
K
=
K
(
α
) is still the splitting
field of
{
f
i
(
x
)
}
over
F
(
α
). Also,
K
(
β
) is the splitting field of
{
f
i
(
x
)
}
over
F
(
β
).
By Theorem 27,
σ
extends to an isomorphism
φ
:
K
→
K
(
β
). In particular, [
K
:
F
] = [
K
(
β
) :
F
], and
K
=
K
(
β
). Therefore
f
splits completely in
K
[
x
].
Conversely, suppose any irreducible polynomial in
F
[
x
] that has a
root in
K
splits completely in
K
[
x
]. Since
K
is a finite extension of
F
,
K
=
F
(
α
1
, α
2
, . . . , α
n
). Let
f
i
be the irreducible polynomial over
F
with
α
i
as a root. Then by assumption,
f
i
splits completely in
K
[
x
],
and
K
contains the splitting field of
{
f
i
}
. But any subfield of
K
would
not contain all the roots of. Therefore
K
is the splitting field of
{
f
i
}
.
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 Spring '09
 SusamaAgarwala
 Algebra, finite field, splitting field

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