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* . [26] 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 discriminant 2√* m* ≈ . [27] If we let * r* and 2* s* be the number of real and complex embeddings, so that * n* = * r* + 2* s* , put * ρ* = * r* /* n* and * σ* = 2* s* /* n* . Set * α* (* ρ* , * σ* ) to be the infimum of rd * K* for * K* with (* r', 2* s') = (* ρn* , * σn* ). We have (for all n large enough) [27]