Tamely Ramified Towers and Discriminant Bounds for Number Fields—II

Abstract

The root discriminant of a number field of degree n is the n th root of the absolute value of its discriminant. Let R 0(2 m) be the minimal root discriminant for totally complex number fields of degree 2 m, and put α 0 = lim inf m R 0(2 m). Define R 1 ( m) to be the minimal root discriminant of totally real number fields of degree m and put α 1 = lim inf m R 1( m). Assuming the Generalized Riemann Hypothesis, α 0 ≥ 8 πe \\gamma ≈ 44.7, and, α 1 ≥ 8 πe \\gamma + π / 2 ≈ 215.3. By constructing number fields of degree 12 with suitable properties, we give the best known upper estimates for α 0and α 1 : α 0 < 82.2, α 1 < 954.3.