Counting $D_4$-field extensions by multi-invariants

TL;DR

This paper counts $D_4$-field extensions over $Q$ using multi-invariants, verifies their asymptotic distribution as predicted by Gundlach's version of Malle's conjecture, and compares constants with recent theoretical predictions.

Contribution

It introduces a counting method for $D_4$-extensions using multi-invariants and confirms the predicted asymptotic behavior, providing evidence for Gundlach's conjecture.

Findings

Confirmed the asymptotic behavior predicted by Gundlach's version of Malle's conjecture.

Compared the leading constant with recent theoretical predictions by Loughran and Santens.

Provided explicit counts of $D_4$-extensions ordered by multi-invariants.

Abstract

We count the number of Galois extensions $M/\mathbb{Q}$ with fixed Galois group $\text{Gal}(M/\mathbb{Q})=D_4$ ordered by multi-invariants introduced by Gundlach. We verify the asymptotic behavior predicted by Gundlach's version of Malle's conjecture. We compare the leading constant to recent predictions by Loughran and Santens.