Order of zeros of Dedekind zeta functions

Abstract

Answering a question of Browkin, we provide a new unconditional proof that the Dedekind zeta function of a number field L L has infinitely many nontrivial zeros of multiplicity at least 2 if L L has a subfield K K for which L / K L/K is a nonabelian Galois extension. We also extend this to zeros of order 3 when G a l ( L / K ) Gal(L/K) has an irreducible representation of degree at least 3, as predicted by the Artin holomorphy conjecture.