On the compositum of all degree d extensions of a number field

TL;DR

This paper investigates the structure of the compositum of all degree d extensions over a number field, establishing conditions under which all smaller degree extensions are contained and when a uniform bound on generators exists, especially for Galois extensions.

Contribution

It characterizes when all smaller degree extensions lie in the compositum and when a uniform generator bound exists, with new results for Galois extensions and prime degrees.

Findings

All degree less than 5 extensions lie in the compositum if and only if d<5.

A uniform generator bound exists if and only if d<3.

For prime d, all Galois subextensions are generated by elements of degree at most d.

Abstract

Let k be a number field, and denote by k^[d] the compositum of all degree d extensions of k in a fixed algebraic closure. We first consider the question of whether all algebraic extensions of k of degree less than d lie in k^[d]. We show that this occurs if and only if d < 5. Secondly, we consider the question of whether there exists a constant c such that if K/k is a finite subextension of k^[d], then K is generated over k by elements of degree at most c. This was previously considered by Checcoli. We show that such a constant exists if and only if d < 3. This question becomes more interesting when one restricts attention to Galois extensions K/k. In this setting, we derive certain divisibility conditions on d under which such a constant does not exist. If d is prime, we prove that all finite Galois subextensions of k^[d] are generated over k by elements of degree at most d.