tower of fields shows that the root discriminant does not change in an unramified extension. Asymptotic lower bounds Given nonnegative rational numbers ρ and σ, not both 0, and a positive integer n such that the pair (r,2s) = (ρn,σn) is in Z × 2Z, let αn(ρ, σ) be the infimum of rdK as K ranges over degree n number fields with r real embeddings and 2s complex embeddings, and let α(ρ, σ) = liminfn→∞ αn(ρ, σ). Then , and the generalized Riemann hypothesis implies the stronger bound There is also a lower bound that holds in all degrees, not just asymptotically: For totally real fields, the root discriminant is > 14, with 1229 exceptions. Asymptotic upper bounds On the other hand, the existence of an infinite class field tower can give upper bounds on the values of α(ρ, σ). For example, the infinite class field tower over Q() with m = 3·5·7·11·19 produces fields of arbitrarily large degree with root discriminant 2 ≈ 296.276, so α(0,1) < 296.276. Using tamely ramified towers, Hajir and Maire have shown that α(1,0) < 954.3 and α(0,1) < 82.2, improving upon earlier bounds of Martinet. Relation to other quantities When embedded into , the volume of the fundamental domain of OK is (sometimes a different measure is used and the volume obtained is , where r2 is the number of complex places of K). Due to its appearance in this volume, the discriminant also appears in the functional equation of the Dedekind zeta function of K, and hence in the analytic class number formula, and the Brauer–Siegel theorem. The relative discriminant of K/L is the Artin conductor of the regular representation of the Galois group of K/L. This provides a relation to the Artin conductors of the characters of the Galois group of K/L, called the conductor-discriminant formula. Notes References Primary sources Secondary sources Further reading Algebraic number theory