THE IDELIC APPROACH TO CLASS GROUPS AND UNIT
GROUPS
PETE L. CLARK
Contents
7.1.
Rings of
S
integers
1
7.2.
Finiteness of the
S
class groups
1
7.3.
Structure of the
S
unit groups
5
7.1.
Rings of
S
integers.
Notation:
Despite some misgivings, we introduce the following notation.
For a
global field
K
, we let Σ
K
denote the set of all places of
K
, i.e., equivalence classes
of nontrivial norms on
K
. (Recall that at this point we have associated a canon
ical normalized Artin absolute value to each place in
K
.) Further we write Σ
NA
K
for the subset of nonArchimeden places of
K
, and let Σ
Arch
K
denote the subset of
Archimedean places of
K
, which is nonempty iff
K
is a number field.
Now let
S
⊂
Σ
K
be a finite set such that
S
⊃
Σ
Arch
K
.
We define the ring
R
S
of
Sintegers
of
K
to be the set of
x
∈
K
such that
v
(
x
)
≥
0 for all
v
∈
S
\
Σ
Arch
K
.
Exercise 7.1:
a) If
K
is a function field and
S
=
∅
, show that
R
S
is equal to the maximal finite
subfield of
K
(or, in other words, the “algebraic closure of
F
p
in
K
”.)
b) Show that if
S
=
∅
, then
R
S
is a Dedekind domain that is not a field.
Exercise 7.2: Suppose that
S
⊂
T
are finite subsets of Σ
K
with Σ
Arch
K
⊂
S
.
a) Show that
R
S
⊂
R
T
.
b)* Show that
R
T
is a localization of
R
S
. (The special case that
R
S
is a PID is
straightforward.
In order to do the general case, it may be necessary to use the
finiteness of the ideal class group of
R
S
, coming up shortly.)
7.2.
Finiteness of the
S
class groups.
Because
R
S
is a Dedekind domain, it has an interesting invariant attached to
it: its ideal class group Cl(
R
S
).
This is the free abelian group generated by the
nonzero prime ideals, which we may view as being in correspondence with the non
Archimedean places
v
of
K
.
Inside Frac(
R
S
) we have the subgroup of principal
fractional ideals (
f
) for
f
∈
K
×
. Explicitly, we view (
f
) as an element of
v
Z
[
v
]
by sending
f
to the sequence
{
v
(
f
)
}
, which has all but finitely many elements 0. Let
1
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PETE L. CLARK
us write this subgroup as Prin(
K
). By definition, the
ideal class group
Cl(
R
S
)
of
R
S
is the quotient Frac(
R
S
)
/
Prin(
R
S
). To summarize, we have exact sequences
1
→
R
×
S
→
K
×
→
Prin(
R
S
)
→
1
,
1
→
Prin(
R
S
)
→
Frac(
R
S
)
→
Cl(
R
S
)
→
1
.
Note that this discussion makes perfect sense for an arbitrary Dedekind domain
R
.
Indeed, a celebrated theorem of L. Claborn asserts that for any abelian group
A
whatsoever, there exists a Dedekind domain
R
such that Cl(
R
)
∼
=
A
. However, in
the arithmetic case, we have the following important result, a generalization of the
second of the three fundamental finiteness theorems in classical algebraic number
theory.
Theorem 1.
Let
K
be a global field and
S
⊂
Σ
K
a finite set containing all the
Archimedean places. Then the ideal class group
Cl(
R
S
) = Frac(
K
)
/
Prin(
K
)
is a
finite abelian group.
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 Fall '11
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 Number Theory, Integers, Abelian group, Algebraic number theory, Ik, ideal class group

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