Arithmetical equivalence of fields and group actions

ZHANG Ji-ping

Abstract:
As is known, prime decomposition in extensions of rational
field is determined by some analytic object, namely Dedekind's
zeta function. Two number fields are said to be arithmetically
equivalent if they have the same Dedekind's zeta function.
 From the group theoretical point of view  arithmetical
equivalence of fields  can be characterized in terms of the
permutation characters of corresponding Galois groups. In this
talk we will  report some progress on this problem.