Discriminant of An Algebraic Number Field - Root Discriminant

The root discriminant of a number field, K, of degree n, often denoted rd_K, is defined as the n-th root of the absolute value of the (absolute) discriminant of K. The relation between relative discriminants in a tower of fields shows that the root discriminant does not change in an unramified extension. The existence of a class field tower provides bounds on the root discriminant: the existence of an infinite class field tower over Q(√-m) where m = 3·5·7·11·19 shows that there are infinitely many fields with root discrimininant 2√m ≈ 296.276. If we let r and 2s be the number of real and complex embeddings, so that n=r+2s, put ρ=r/n and σ=2s/n. Set α(ρ,σ) to be the infimum of rd_K for K with (r',2s') = (ρn,σn). We have

and on the assumption of the Generalized Riemann hypothesis

So we have α(0,1) < 296.276. Martinet has shown α(0,1) < 93 and α(1,0) < 1059. Voight 2008 proves that for totally real fields, the root discriminant is > 14, with 1229 exceptions.