Mathematical Research Letters
Volume 25 (2018)
Counting $G$-extensions by discriminant
Pages: 1151 – 1172
The problem of analyzing the number of number field extensions $L/K$ with bounded (relative) discriminant has been the subject of renewed interest in recent years, with significant advances made by Schmidt, Ellenberg–Venkatesh, Bhargava, Bhargava–Shankar–Wang, and others. In this paper, we use the geometry of numbers and invariant theory of finite groups, in a manner similar to Ellenberg and Venkatesh, to give an upper bound on the number of extensions $L/K$ with fixed degree, bounded relative discriminant, and specified Galois closure.
discriminants, number field counting, $G$-extensions, discriminant counting, polynomial invariant theory, geometry of numbers
2010 Mathematics Subject Classification
Primary 11R21. Secondary 11H06, 11R29, 13A50.
Received 29 October 2016
Accepted 3 April 2017
Published 16 November 2018