The density of discriminants of quartic rings and fields

By Manjul Bhargava. 2005

315 citations recorded by the source corpus.

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 interpretation of the various constants appearing in these theorems in terms of local masses of quintic rings and fields.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/217d2d6f44dca02bd4cbb22fc11d5e0ffde4efef

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Citations in the corpus

The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point (2012, methodology): “This gives an injective map from the set of rational orbits of SO(W ) on V with characteristic polynomial f(x) to the set of elements in the Galois cohomology group H(Q, J [2]), which classifies principal homogeneous spaces for this finite group scheme.”
A construction of polynomials with squarefree discriminants (2011, methodology): “This is far inferior to what can be shown using other methods: for n ≤ 5 the number of such fields is known to be O(N) by work of Bhargava [2, 3], while for larger n, Ellenberg and Venkatesh [5] have given a lower bound of cN 2 .”
The number of extensions of a number field with fixed degree and bounded discriminant (2003, background): “This conjecture is trivial when n = 2; it has been proved for n = 3 by Davenport and Heilbronn [7] in case K = Q, and by Datskovsky and Wright in general [6]; and for n = 4, 5 and K = Q by Bhargava [3], [2].”
Geometry-of-numbers methods over global fields I: Prehomogeneous vector spaces (2015, unknown): “The cases n = 4 and 5 for F = Q were carried out in [8, 9].”
Families of L -Functions and Their Symmetry (2014, background): “The theory over Z as needed to determine the density of such quartic and quintic fields is due to Bhargava ([10], [11]).”
Higher composition laws IV: The parametrization of quintic rings (2008, background): “Every quintic ring has at least one sextic resolvent ring.”
On the Davenport–Heilbronn theorems and second order terms (2010, unknown): “, [3], [4], [6], [11], [17], [18], [27], [28]).”
Secondary terms in counting functions for cubic fields (2011, background): “The work of Belabas, Bhargava, and Pomerance.”