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(2 m) be the minimal root discriminant for totally complex number fields of degree 2 m, and put α0 = mR0(2 m). DefineR1 (m) to be the minimal root discriminant of totally real number fields of degree m and put α1 = mR1(m). Assuming the Generalized Riemann Hypothesis, α0 ≥ 8πe\gamma ≈ 44.7, and,α1 ≥ 8πe\gamma + π / 2 ≈ 215.3. By constructing number fields of degree 12 with suitable properties, we give the best known upper estimates for α0andα1 :α0 < 82.2,α1 < 954.3.