The density of discriminants of quartic rings and fields

@article{Bhargava2005TheDO,
  title={The density of discriminants of quartic rings and fields},
  author={M. Bhargava},
  journal={Annals of Mathematics},
  year={2005},
  volume={162},
  pages={1031-1063}
}
  • M. Bhargava
  • Published 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… Expand
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