The root discriminant of a number field of degree n is the nth root of the absolute value of its discriminant. Let R_0(2m)be the minimal roo discriminant for totally complex number fields of degree 2m, and putα_0=lim inf_m R_0(2m). define R_1(m)to be the minimal root discriminant of totally real number fields of degree m and putα_1= lim inf_m R_1(m). assuming the Generalized Riemann Hypothesisα_0≥8πe~γ≈44.7, and, α_1≥8πe~γ+π/2≈215.3. By constructing number fields of degree 12 with suitable properties, we give the best known upper estimates forα_0 andα_0<82.2,α_1<954.3.
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