The number of extensions of a number field with fixed degree and bounded discriminant

TL;DR

This paper establishes improved upper bounds on the number of degree-specific extensions of a fixed number field with discriminant below a certain threshold, and also provides related lower bounds and bounds for Galois extensions.

Contribution

It introduces sharper upper bounds on the count of such extensions, advancing previous results by Schmidt, and explores related bounds for Galois extensions.

Findings

Improved upper bounds on the number of extensions with fixed degree and bounded discriminant.

Lower bounds for the number of such extensions.

Upper bounds specifically for Galois extensions.

Abstract

We give an upper bound on the number of extensions of a fixed number field of prescribed degree and discriminant less than X; these bounds improve on work of Schmidt. We also prove various related results, such as lower bounds for the number of extensions and upper bounds for Galois extensions.