Editing Algebraic number theory (section)

===Zeta function===
The [[Dedekind zeta function]] of a number field, analogous to the [[Riemann zeta function]], is an analytic object which describes the behavior of prime ideals in {{math|''K''}}. When {{math|''K''}} is an abelian extension of {{math|'''Q'''}}, Dedekind zeta functions are products of [[Dirichlet L-function]]s, with there being one factor for each [[Dirichlet character]]. The trivial character corresponds to the Riemann zeta function. When {{math|''K''}} is a [[Galois extension]], the Dedekind zeta function is the [[Artin L-function]] of the [[regular representation]] of the Galois group of {{math|''K''}}, and it has a factorization in terms of irreducible [[Artin representation]]s of the Galois group.

The zeta function is related to the other invariants described above by the [[class number formula]].