The zeros of Dedekind zeta functions and class numbers of CM-fields

Abstract

Let F'/F be a finite normal extension of number fields with Galois group Gal(F'/F). Let. be an irreducible character of Gal(F'/F) of degree greater than one and L(s,chi) the associated Artin L-function. Assuming the truth of Artin's conjecture, we have explicitly determined a zero-free region about 1 for L( s,.). As an application we show that, for a CM-field K of degree 2n with solvable normal closure over Q, if n >= 370 as well as n is not an element of {384, 400, 416, 448, 512}, then the relative class number of K is greater than one.