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# 3 Transcendental computation of the Class number

We first need to introduce the Dedekind zeta function for a number field K, and its Euler product expansion

(s) = =

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

(s - 1) =

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 I0 C, this latter set is bijective to the set {aR I0-1 | N(a) r . N(I0)-1}. (Here N(a) denotes the modulus of the norm of a.) We have a natural embedding K K R. The image of J = I0-1 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 I0-1 | 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 ) rd} 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

(K R/J) = N(J)(K R/R).

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 r2 of K.

(K R/R) = .

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) = (Frobq) is the value of on a Frobenius element associated with q. We define the Dirichlet L-series and their Euler product formulas as follows

L(s,) = =

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 fp denotes the residue field extension over p and gp the number of distinct primes in K lying over p. The product expansion of (s) becomes

(s) = F(s) . L(s,).

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

(s - 1)(s) = L(1,).

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

L(s,) = (x) .

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

L(s,) = (x) .

The expression

() = (x)

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

= - log(1 - )

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