Solvable Number Field Extensions of Bounded Root Discriminant

TL;DR

This paper proves that for any number field, there are only finitely many solvable extensions of a given degree with bounded root discriminant, advancing understanding of class field towers and discriminant bounds.

Contribution

It establishes a finiteness result for solvable number field extensions with bounded root discriminant, a new insight in algebraic number theory.

Findings

Finiteness of solvable extensions with bounded root discriminant

Applicable to class field tower studies

Provides bounds on extension degrees

Abstract

Let $K$ be a number field and $d_K$ the absolute value of the discrimant of $K/\mathbb{Q}$. We consider the root discriminant $d_L^{\frac{1}{[L:\mathbb{Q}]}}$ of extensions $L/K$. We show that for any $N>0$ and any positive integer n, the set of length n solvable extensions of $K$ with root discriminant less than $N$ is finite. The result is motivated by the study of class field towers.