THE STRONG ARTIN CONJECTURE AND LARGE CLASS NUMBERS

Abstract

Assuming the Generalized Riemann Hypothesis (GRH) and the Artin conjecture for Artin L-functions, Duke found an upper bound of the class number of a totally real field of degree n whose normal closure is an S-n Galois extension over Q. Again under the GRH and the Artin conjecture, he constructed totally real number fields whose Galois closures are S-n with the largest possible class numbers up to a constant. We prove that the strong Artin conjecture is enough to obtain Duke's result. Moreover, we prove the strong Artin conjecture for S-4 and A(4) Galois extensions; hence the case n = 4 is unconditional