ABSTRACT
We limit ourselves to the standard case where the ground field is a number field. For the higher dimensional case, see §9.3.
Let K be a number field, i.e. a finite extension of Q, and let OK be the ring of integers of K. Let VK the set of non-archimedean places of K. The elements of VK correspond to the maximal ideals of OK ; one may identify VK with MaxOK . We denote by pv the maximal ideal corresponding to v ∈ VK , and by |v| the norm of pv (also called the norm of v), i.e. the number of elements of the residue field κ(v) = OK/pv. If x is a real number, we put :
piK(x) = number of v ∈ VK with |v| 6 x. When K = Q, VK is the set of prime numbers, and we have
piK(x) = pi(x) = number of primes p with p 6 x.
