9.1. FINITE AND ALGEBRAIC EXTENSIONS
421
finite field extension, by Proposition
7.3.9
. Proposition
7.3.1
implies that
K
K.a
0
; : : : ; a
n
; b/
is a finite field extension.
It then follows from
Proposition
7.3.4
that
K.a
0
; : : : ; a
n
; b/
is algebraic over
K
, so, in particu
lar,
b
is algebraic over
K
.
Part (b) is a consequence of part (a).
n
Definition 9.1.2.
Suppose that
K
L
is a field extension and that
E
and
F
are intermediate fields,
K
E
L
and
K
F
L
. The
composite
E
F
of
E
and
F
is the smallest subfield of
L
containing
E
and
F
.
F
E
F
L
[j
[j
K
E
Proposition 9.1.3.
Suppose that
K
L
is a field extension and that
E
and
F
are intermediate fields,
K
E
L
, and
K
F
L
.
(a)
If
E
is algebraic over
K
and
F
is arbitrary, then
E
F
is alge
braic over
F
.
(b)
If
E
and
F
are both algebraic over
K
, then
E
F
is algebraic
over
K
.
(c)
dim
F
.E
F /
dim
K
.E/
.
Proof.
Exercises
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 Fall '08
 EVERAGE
 Logic, Algebra, Prove Proposition, Field extension, finite field extension

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