Tamely Ramified Towers and Discriminant Bounds for Number Fields-II

The root discriminant of a number field of degree n is the nth root of the absolute value of its discriminant. Let R0(2m) be the minimal root discriminant for totally complex number fields of degree 2m, and put α0 = lim infmR0(2m). Define R1(m) to be the minimal root discriminant of totally real number fields of degree m and put α1 = lim infmR1(m). Assuming the Generalized Riemann Hypothesis, α0 ≥ 8πeγ ≈ 44.7, and, α1 ≥ 8πeγ+π/2 ≈ 215.3. By constructing number fields of degree 12 with suitable properties, we give the best known upper estimates for α0 and α1: α0 < 82.2, α1 < 954.3. c © 2002 Elsevier Science Ltd