Note that
OEB
W
kŁ
OEB=
T
p
i
W
kŁ
D
P
OEB=
p
i
W
kŁ
r:
Therefore
B
has only finitely many prime ideals, say
p
1
;:::;
p
g
where
g
OEB
W
kŁ
, and
T
p
i
D
0
. When we take
r
D
g
in (*) we
107

find that
B
D
Y
g
i
D
1
B=
p
i
:
For each
i
,
B=
p
i
is a field, and it is a finite extension of
k
.
Because
k
is perfect, it is even a separable extension of
k
. Now
we can apply (2.26) to deduce that disc
..B=
p
i
/=k/
¤
0
, and
we can apply the preceding lemma to deduce that disc
.B=k/
¤
0
.
2
We now prove the theorem. From the first lemma, we see
that
disc
.B=A/
mod
p
D
disc
..B=
p
B/=.A=
p
//;
and from the last lemma that disc
..B=
p
B/=.A=
p
//
D
0
if
and
only
B=
p
B
is
not
reduced.
Let
p
B
D
Q
P
e
i
i
.
Then
B=
p
B
'
Q
B=
P
e
i
, and
Q
B=
P
e
i
is reduced
”
each
B=
P
e
i
is reduced
”
each
e
i
D
1:
R
EMARK
3.39 (a) In fact there is a precise, but complicated,
relation between the power of
p
dividing disc
.B=A/
and the
extent to which
p
ramifies in
B
. It implies for example that
ord
p
.
disc
.B=A//
P
f
i
.e
i
1/
, and that equality holds if no
e
i
is divisible by the characteristic of
A=
p
. See Serre 1962, III
6.
(b) Let
A
be the ring of integers in a number field
K
, and
let
B
be the integral closure of
A
in a finite extension
L
of
K
.
It is possible to define disc
.B=A/
as an ideal without assuming
B
to be a free
A
-module. Let
p
be an ideal in
A
, and let
S
D
108

A
p
. Then
S
1
A
D
A
p
is principal, and so we can define
disc
.S
1
B=S
1
A/
. It is a power
.
p
A
p
/
m.
p
/
of
p
A
p
. Define
disc
.B=A/
D
Y
p
m.
p
/
:
The index
m.
p
/
is nonzero for only finitely many
p
, and so this
formula does define an ideal in
A
. Clearly this definition agrees
with the usual one when
B
is a free
A
-module, and the above
proof shows that a prime ideal
p
ramifies in
B
if and only if it
divides disc
.B=A/:
E
XAMPLE
3.40 (For experts on Riemann surfaces.) Let
X
and
Y
be compact connected Riemann surfaces, and let
˛
W
Y
!
X
be a nonconstant holomorphic mapping. Write
M
.X/
and
M
.Y /
for the fields of meromorphic functions on
X
and
Y
.
The map
f
7!
f
ı
˛
is an inclusion
M
.X/ ,
!
M
.Y /
which
makes
M
.Y /
into a field of finite degree over
M
.X/
; let
m
be this degree. Geometrically, the map is
m
W
1
except at a finite
number of branch points.
Let
P
2
X
and let
O
P
be the set of meromorphic functions
on
X
that are holomorphic at
P
— it is the discrete valua-
tion ring attached to the discrete valuation ord
P
, and its max-
imal ideal is the set of meromorphic functions on
X
that are
zero at
P
. Let
B
be the integral closure of
O
P
in
M
.Y /
. Let
˛
1
.P /
D f
Q
1
;:::;Q
g
g
and let
e
i
be the number of sheets of
Y
over
X
that coincide at
Q
i
. Then
p
B
D
Q
q
e
i
i
where
q
i
is
the prime ideal
f
f
2
B
j
f .Q
i
/
D
0
g
:
109

Finding factorizations
The following result often makes it very easy to factor an ideal
in an extension field. Again
A
is a Dedekind domain with field
of fractions
K
, and
B
is the integral closure of
A
in a finite
separable extension
L
of
K
.

#### You've reached the end of your free preview.

Want to read all 327 pages?

- Summer '13
- aklal
- Algebra, Number Theory, The Land, Ring, Prime number, Algebraic number theory, Integral domain, Principal ideal domain