26
LECTURE 4. IRREDUCIBLE ALGEBRAIC SETS AND RATIONAL FUNCTIONS
We have
P
(
T
1
, . . . , T
n
)
≡
0
on
V
\
Z
.
Since
V
\
Z
is dense in the Zariski
topology,
P
≡
0
on
V
, i.e.
P
∈
I
(
V
)
. This shows that under the map
R
(
W
)
→
R
(
V
)
,
F
goes to
0
. Since the homomorphism
R
(
W
)
→
R
(
V
)
is injective (any
homomorphism of fields is injective) this is absurd.
In particular, taking
W
=
A
1
k
(
K
)
, we obtain the interpretation of elements
of the field
R
(
V
)
as nonconstant rational functions
V

→
K
defined on an
open subset of
V
(the complement of the set of the zeroes of the denominator).
From this point of view, the homomorphism
R
(
W
)
→
R
(
V
)
defining a rational
map
f
:
V
 →
W
can be interpreted as the homomorphism
f
*
defined by the
composition
φ
→
φ
◦
f
.
Definition 4.4.
A rational map
f
:
V
 →
W
is called
birational
if the cor
responding field homomorphism
f
*
:
R
(
W
)
→
R
(
V
)
is an isomorphism. Two
irreducible affine algebraic sets
V
and
W
are said to be
birationally isomorphic
if there exists a birational map from
V
to
W
.
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 Spring '09
 wei
 Algebra, Geometry, Topology, Rational Functions, Sets, Category theory, Morphism, Zariski, birational map

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