• Corpus ID: 119127997

Statistics of genus numbers of cubic fields

  title={Statistics of genus numbers of cubic fields},
  author={Kevin J. McGown and Amanda Tucker},
  journal={arXiv: Number Theory},
We prove that approximately $96.23\%$ of cubic fields, ordered by discriminant, have genus number one, and we compute the exact proportion of cubic fields with a given genus number. We also compute the average genus number. Finally, we show that a positive proportion of totally real cubic fields with genus number one fail to be norm-Euclidean. 
4 Citations
Counting quintic fields with genus number one
It is proved that a positive proportion of quintic fields have arbitrarily large genus number; and the average genus number of quintIC fields is the average of genus number one and two.
Improved error estimates for the Davenport-Heilbronn theorems
We improve the error terms in the Davenport–Heilbronn theorems on counting cubic fields to O(X). This improves on separate and independent results of the authors and Shankar and Tsimerman [BST13,
The average genus for bouquets of circles and dipoles
The bouquet of circles $B_n$ and dipole graph $D_n$ are two important classes of graphs in topological graph theory. For $n\geq 1$, we give an explicit formula for the average genus


Secondary terms in counting functions for cubic fields
We prove the existence of secondary terms of order X^{5/6} in the Davenport-Heilbronn theorems on cubic fields and 3-torsion in class groups of quadratic fields. For cubic fields this confirms a
The density of abelian cubic fields
In the following note we show that the abelian cubic fields are rare in relation to all cubic fields over the rationals. This is no surprise since an irreducible cubic equation generates an abelian
Let K be a cyclic number eld of prime degree '. Heilbronn showed that for a given ' there are only nitely many such elds that are norm- Euclidean. In the case of ' = 2 all such norm-Euclidean elds
Density of cubic field discriminants
A conjectural refinement of the Davenport- Heilbronn theorem on the density of cubic field discriminants is given, which is plausible theoretically and agrees very well with computa- tional data.
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.
The genus field and genus group in finite number fields, II
In the preceding paper of the same title (cf. [1]) I defined the notion of the principal genus G K of a finite number field K as the least ideal group, which contains the group I K of totally
This article, which is an update of a version published 1995 in Expo. Math., intends to survey what is known about Euclidean number fields; we will do this from a number theoretical (and number
Error estimates for the Davenport-Heilbronn theorems
We obtain the first known power-saving remainder terms for the theorems of Davenport and Heilbronn on the density of discriminants of cubic fields and the mean number of 3-torsion elements in the
The genus fields of algebraic number fields
{REPLACEMENT-(...)-( )} A simple construction of genus fields of abelian number fields The narrow genus field K of k (genus field of k in the sense of Fröhlich) is defined as the . The genus field
A fast algorithm to compute cubic fields
  • K. Belabas
  • Computer Science, Physics
    Math. Comput.
  • 1997
A very fast algorithm to build up tables of cubic fields with discriminant up to 10 11 and complex cubic fields down to -10 11 has been computed.