Upper bound on the number of extensions of a given number field

2020/10/26 by Lee, Jungin
#11F06 #11H06 #11R04 #11R21 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2010.13489

Abstract

In this paper we improve the upper bound of the number N_{K, n}(X) of degree n extensions of a number field K with absolute discriminant bounded by X. This is achieved by giving a short O_K-basis of an order of an extension L of K. Our result generalizes the best known upper bound on N_{ℚ, n}(X) by Lemke Oliver and Thorne to all number fields K. Precisely, we prove that N_{K, n}(X) ≪_{K, n} X^{c (log n)2} for an explicit constant c independent on K and n. We also improve the upper bound of the number of maximal arithmetic subgroups in certain connected semisimple Lie groups.

Related

• The number of extensions of a number field with fixed degree and bounded discriminant
• Counting G-Extensions by Discriminant
• Upper bounds on number fields of given degree and bounded discriminant
• Enumerating Galois extensions of number fields
• Uniform exponent bounds on the number of primitive extensions of number fields