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7.3. ADJOINING ALGEBRAIC ELEMENTS TO A FIELD
325
(c)
dim
K
.K.˛//
D
deg
.f /
.
Proof.
We have shown that the ring of polynomials in
˛
with coefﬁcients
in
K
is a ﬁeld, isomorphic to
KŒxŁ=.f.x//
. Therefore,
K.˛/
D
KŒ˛Ł
Š
KŒxŁ=.f.x//
. This shows part (a). For any
p
2
KŒxŁ
, write
p
D
qf
C
r
,
where
r
D
0
or deg
.r/ <
deg
.f /
. Then
p.˛/
D
r.˛/
, since
f.˛/
D
0
.
This means that
f
1;˛;:::;˛
d
±
1
g
spans
K.˛/
over
K
. But this set is also
linearly independent over
K
, because
˛
is not a solution to any equation
of degree less than
d
. This shows parts (b) and (c).
n
Example 7.3.7.
Consider
f.x/
D
x
2
±
2
2
Q
.x/
, which is irreducible by
Eisenstein’s criterion (Proposition
6.8.4
). The element
p
2
2
R
is a root
of
f.x/
. (The existence of this root in
R
is a fact of
analysis
.) The ﬁeld
Q
.
p
2/
Š
Q
ŒxŁ=.x
2
±
2/
consists of elements of the form
a
C
b
p
2
, with
a;b
2
Q
. The rule for addition in
Q
.
p
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 Fall '08
 EVERAGE
 Algebra, Polynomials

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