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438
9. FIELD EXTENSIONS – SECOND LOOK
The following is an important technical result that is used in the sequel.
Proposition 9.5.6.
Let
K
±
L
be a ﬁnite–dimensional ﬁeld extension and
let
A;B
be ﬁelds intermediate between
K
and
L
. Suppose that
B
is Galois
over
K
. Then
A
²
B
is Galois over
A
and
Aut
A
.A
²
B/
Š
Aut
A
\
B
.B/
.
Proof.
Exercise
9.5.2
.
n
The remainder of this section can be omitted without loss of continuity.
It is not hard to obtain the inequality of Corollary
9.4.18
without the
separability assumption. It follows that the separability assumption can
also be removed in Proposition
9.5.3
. Although we consider only separable
ﬁeld extensions in this text, the argument is nevertheless worth knowing.
Proposition 9.5.7.
Let
L
be a ﬁeld. Any collection of distinct automor
phisms of
L
is linearly independent.
Proof.
Let
f
±
1
;:::;±
n
g
be a collection of distinct automorphisms of
L
.
We show by induction on
n
that the collection is linearly independent (in
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 Fall '08
 EVERAGE
 Algebra

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