Number Theory: An Introduction to Mathematics

386 IX The Number of Prime Numbers

whereAais a positive constant which can be explicitly described. (The expression for
Aawhich Artin conjectured requires modification in some cases.)
There are also analogues for function fields of these results for number fields. Let
Kbe an arbitrary field. Afield of algebraic functions of one variable over Kis a field
Lwhich satisfies the following conditions:

(i) K⊆L,

(ii) Lcontains an elementυwhich istranscendentaloverK,i.e.υsatisfies no monic

polynomial equation

un+a 1 un−^1 +···+an= 0

with coefficientsaj∈K,
(iii)Lis afinite extensionof the fieldK(υ)of rational functions ofυwith coeffients
fromK,i.e.Lis finite-dimensional as a vector space overK(υ).
LetRbe a ring withK⊆R⊂Lsuch thatx∈L\Rimpliesx−^1 ∈R. Then the
setPof alla∈Rsuch thata=0ora−^1 ∈/Ris an ideal ofR, and actually the unique
maximal ideal ofR. Hence the quotient ringR/Pis a field. SinceRis the set of all
x∈Lsuch thatxP⊆P, it is uniquely determined byP. The idealPwill be called a
prime divisorof the fieldLandR/Pitsresidue field. It may be shown that the residue
fieldR/Pis a finite extension of (a field isomorphic to)K.
An arbitrarydivisorof the fieldLis a formal productA=

∏

PP

vPover all prime

divisorsPofL, where the exponentsvPare integers only finitely many of which are
nonzero. The divisor isintegralifvP≥0forallP.
The setK′of all elements ofLwhich satisfy monic polynomial equations with co-
efficients fromKis a subfield containingK,andLis also a field of algebraic functions
of one variable overK′. It is easily shown that no element ofL\Rsatisfies a monic
polynomial equation with coefficients fromR. ConsequentlyK′⊆Rand the notion
of prime divisor is the same whether we considerLto be overKor overK′.Since
(K′)′=K′, we may assume from the outset thatK′=K. The elements ofKwill
then be calledconstantsand the elements ofL functions.
Suppose now that the field of constantsKis a finite fieldFqcontainingqelements.
We define thenorm N(P)of a prime divisorPto be the cardinality of the associated
residue fieldR/Pand the norm of an integral divisorA=

∏

PP

vPto be

N(A)=

∏

P

N(P)vP.

It may be shown that, for each positive integerm, there exist only finitely many prime
divisors of normqm. Moreover, forRs>1 the zeta-function ofLcan be defined by

ζL(s)=

∑

A

N(A)−s,

where the sum is over all integral divisors ofL,andthen

ζL(s)=

∏

P

( 1 −N(P)−s)−^1 ,

where the product is over all prime divisors ofL.