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Computational data on ${\mathfrak S}_n$-extensions of $\mathbb Q$

Gunter Malle

Abstract

We discuss computational results on field extensions $K/{\mathbb Q}$ of degree $n\le11$ with Galois group of the Galois closure isomorphic to the full symmetric group ${\mathfrak S}_n$. More precisely, we present statistics on the number of such extensions as a function of the field discriminant and compare them to the known predictions by Bhargava and the author. We also investigate the numbers of fields with equal discriminant and tabulate class numbers and class groups to compare them against Cohen--Lenstra--Martinet type of heuristics and their proposed improvements.

Computational data on ${\mathfrak S}_n$-extensions of $\mathbb Q$

Abstract

We discuss computational results on field extensions of degree with Galois group of the Galois closure isomorphic to the full symmetric group . More precisely, we present statistics on the number of such extensions as a function of the field discriminant and compare them to the known predictions by Bhargava and the author. We also investigate the numbers of fields with equal discriminant and tabulate class numbers and class groups to compare them against Cohen--Lenstra--Martinet type of heuristics and their proposed improvements.
Paper Structure (18 sections, 13 theorems, 15 equations, 30 tables)

This paper contains 18 sections, 13 theorems, 15 equations, 30 tables.

Table of Contents

  1. Enumeration of ${\mathfrak{S}}_n$-extensions with at most one real embedding
  2. Growth rate and Bhargava's constant
  3. Totally complex ${\mathfrak{S}}_4$-extensions
  4. The fields
  5. Discriminant multiplicities
  6. Class groups statistics
  7. Extensions of degree 5 with one real embedding
  8. The fields
  9. Class groups statistics
  10. Totally complex ${\mathfrak{S}}_6$-extensions
  11. The fields
  12. Class groups statistics
  13. Extensions of degree 7 with one real embedding
  14. Totally complex ${\mathfrak{S}}_8$-extensions
  15. Extensions of degree 9 with one real embedding
  16. ...and 3 more sections

Key Result

Proposition 2.1

There are $63\,748\,067$ totally complex number fields of degree $4$ (up to isomorphism) with discriminant at most $10^9$ and Galois group ${\mathfrak{S}}_4$.

Theorems & Definitions (13)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 5.1
  • Proposition 6.1
  • ...and 3 more