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We first need to introduce the Dedekind zeta function for a
number field *K*, and its Euler product expansion
where the sum runs over all ideals *I* of *R* and the product runs
over all prime ideals *Q* of *R*. The two expressions give us two ways
of computing
(*s* - 1)(*s*). The left-hand side is
expressed in terms of ``arithmetic'' invariants and the right-hand
side in terms of invariants for the Galois group. The resulting
identity gives a way for computing the Class number *h* of *K*.
The left-hand limit can be computed to be
The set
{*I* | *N*(*I*) *r*} can be split according to ideal
classes. We try to compute for each ideal class *C*,
*z*(

*C*) =

.

Fixing an ideal *I*_{0} *C*, this latter
set is bijective to the set
{*aR* *I*_{0}^{-1} | *N*(*a*) *r*^{ . }*N*(*I*_{0})^{-1}}.
(Here *N*(*a*) denotes the modulus of the norm of *a*.)
We have a natural embedding
*K* *K* *R*. The image
of
*J* = *I*_{0}^{-1} is a lattice in
*K* *R*. Let
denote the image of *J* - {0} in the quotient
= (*K* *R*)^{*}/*U* where *U* is the image of the group
of units in *R* under the above embedding. There is a natural
homomorphism
*N* : *R*^{*} which restricts to the modulus of the
norm on the image of *K*. We obtain a natural bijection between
{*aR* *I*_{0}^{-1} | *N*(*a*) *r*}
and
{*l* | *N*(*l* ) *r*}. Let denote the
image of
(1/*r*)*J* - {0} in , then we have a natural bijection
between
{*l* | *N*(*l* ) *r*^{d}}
and
{*l* | *N*(*l* ) 1},
where *d* denotes the degree of *K* oover *Q*.
Let
_{ 1} denote locus of
*l* such that
*N*(*l* ) 1. Let denote the Haar measure
on
*K* *R*. This is invariant under the action of *U*
and thus gives a measure also denoted by on . Since *J* is
a lattice in
*K* *R* we have
Moreover, the denominator can be re-written
In particular, we see that the limit *z*(*C*) is independent of the
class *C*. Let
(*K* *R*)^{*}_{1} denote the kernel of the
norm map. This is a group and thus we have a Haar measure on
it. One shows that
(

_{ 1}) =

((

*K* *R*)

^{*}_{1}/

*U*)

Combining the above calculations one obtains
(

*s* - 1)

^{ . }(

*s*) =

*h*^{ . }
This often called the ``Class number formula'' for *K*. Note that the
denominator can be computed in closed form in terms of the
discriminant *D* of the field *K* and the number of pairs of
conjugate complex embeddings *r*_{2} of *K*.
However, the numerator is in general
more complicated since it involves computing the group of units of
*K*.
To expand the right-hand term we restrict our attention to abelian
extensions *K* of *Q*. The product term on the left can be first
grouped according to rational primes
Now for each rational prime *q* which is unramified in *K* we have
where runs over all characters of the Galois group and
(*q*) = (Frob_{q}) is the value of on a Frobenius element
associated with *q*. We define the Dirichlet *L*-series and their
Euler product formulas as follows
where we set (*p*) = 0 when is ramified at *p*. We also
define the additional factor
*F*(

*s*) =

where the product runs over all ramified primes and *f*_{p} denotes the
residue field extension over *p* and *g*_{p} the number of distinct
primes in *K* lying over *p*. The product expansion of
(*s*)
becomes
Thus the computation of the limit can be reduced to the corresponding
computation for the Dirichlet *L*-series. For the case of the unit
character we get by comparison with the zeta function
(

*s* - 1)

*F*(

*s*)

*L*(

*s*, 1) = 1.

So the right-hand limit gives
There is a positive integer *m* such that is determined
on classes modulo *m* and is zero on all primes *p* dividing
it; *m* is called the *conductor* of . We rewrite the
*L*-function associated with as follows
The latter sum can be rewritten using the identity
where is a primitive *m*-th root of unity. The second sum
then becomes
Thus we obtain
The expression
is called the Gaussian sum associated with the integer *i* and the
character . If is not the unit character then
() = 0. Moreover, if *i* 0 then we have the identity
Hence, we obtain the formula when is not the unit character
*L*(1,

) = -

(

)

^{ . }log(1 -

)

** Next:** 4 Divisiblity of the
** Up:** Kummer's proof of Fermat's
** Previous:** Case 2: p| XYZ
Kapil Hari Paranjape
2002-11-22