Discriminant function and steroids

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]

Discriminant function and steroids

discriminant function and steroids

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