324
7. FIELD EXTENSIONS – FIRST LOOK
Since
K
¤
L
, there exists an element
a
1
2
L
n
K
.
Then
K
K.a
1
/
L
, dim
K
.K.a
1
// > 1
, and
dim
K.a
1
/
.L/
D
dim
K
.L/=
dim
K
.K.a
1
// <
dim
K
.L/:
By the induction hypothesis applied to the pair
K.a
1
/
L
, there ex
ists a natural number
n
and there exist finitely many elements
a
2
; : : : ; a
n
,
such that
L
D
K.a
1
/.a
2
; : : : ; a
n
/
D
K.a
1
; a
2
; : : : ; a
n
/:
This completes
the proof.
n
The set
I
˛
of polynomials
p.x/
2
KOExŁ
satisfying
p.˛/
D
0
is
the kernel of the homomorphism
'
˛
W
KOExŁ
!
L
, so is an ideal in
KOExŁ
.
We have
I
˛
D
f .x/KOExŁ
, where
f .x/
is an element of mini
mum degree in
I
˛
, according to Proposition
6.2.27
. The polynomial
f .x/
is necessarily irreducible; if it factored as
f .x/
D
f
1
.x/f
2
.x/
, where
deg
.f / >
deg
.f
i
/ > 0
, then
0
D
f .˛/
D
f
1
.˛/f
2
.˛/
. But then one
of the
f
i
would have to be in
I
˛
, while deg
.f
i
/ <
deg
.f /
, a contra
diction.
The generator
f .x/
of
I
˛
is unique up to multiplication by a
nonzero element of
K
, so there is a unique monic polynomial
f .x/
such
that
I
˛
D
f .x/KOExŁ
, called the
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 Fall '08
 EVERAGE
 Algebra, minimal polynomial, dimK .K.a1

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