Computational data on \mathfrak S_n-extensions of \mathbb Q

2024/03/14 by Gunter Malle, Malle, Gunter
Mathematics · Computer Science · #Advanced Topology and Set Theory #Coding theory and cryptography #Analytic Number Theory Research

paper · pdf · doi:10.48550/arxiv.2403.09146

Abstract

We discuss computational results on field extensions K/\mathbb Q of degree n≤11 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.

Related

• Counting G-Extensions by Discriminant
• Counting C₂ \wr S₄ fields with a power saving error term
• Combinatorial Techniques in the Galois Theory of p-Extensions
• Counting extensions of number fields with Frobenius Galois group
• Enumerating Galois extensions of number fields