Tamely Ramified Towers and Discriminant Bounds for Number Fields

Abstract

The root discriminant of a number field of degree n is the nth root of the absolute value of its discriminant. Let R_2m be the minimal root discriminant for totally complex number fields of degree 2m, and put α₀ = lim inf_m R_2m. One knows that α₀ [ges ] 4πe^γ ≈ 22.3, and, assuming the Generalized Riemann Hypothesis, α₀ [ges ] 8πe^γ ≈ 44.7. It is of great interest to know if the latter bound is sharp. In 1978, Martinet constructed an infinite unramified tower of totally complex number fields with small constant root discriminant, demonstrating that α₀ < 92.4. For over twenty years, this estimate has not been improved. We introduce two new ideas for bounding asymptotically minimal root discriminants, namely, (1) we allow tame ramification in the tower, and (2) we allow the fields at the bottom of the tower to have large Galois closure. These new ideas allow us to obtain the better estimate α₀ < 83.9.