The density of discriminants of quartic rings and fields

  title={The density of discriminants of quartic rings and fields},
  author={Manjul Bhargava},
  journal={Annals of Mathematics},
  • M. Bhargava
  • Published 1 September 2005
  • Mathematics
  • Annals of Mathematics
We determine, asymptotically, the number of quintic fields having bounded discriminant. Specifically, we prove that the asymptotic number of quintic fields having absolute discriminant at most X is a constant times X. In contrast with the quartic case, we also show that a density of 100% of quintic fields, when ordered by absolute discriminant, have Galois closure with full Galois group $S_5$. The analogues of these results are also proven for orders in quintic fields. Finally, we give an… 
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On the density of discriminants of cubic fields. II
  • H. Davenport, H. Heilbronn
  • Mathematics
    Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
  • 1971
An asymptotic formula is proved for the number of cubic fields of discriminant δ in 0 < δ < X; and in - X < δ < 0.