On number fields with nontrivial subfields

TL;DR

This paper investigates the probability of a number field having nontrivial subfields under different enumeration methods, showing it is zero when ordered by generator height for degrees greater than 6.

Contribution

It provides an asymptotic estimate for the count of algebraic numbers generating fields with specified subfields, clarifying the impact of enumeration method on probability.

Findings

Probability is zero when fields are ordered by generator height for degrees > 6.

Provides asymptotic formulas for algebraic numbers generating fields with given subfields.

Highlights the dependence of subfield probability on enumeration method.

Abstract

What is the probability for a number field of composite degree $d$ to have a nontrivial subfield? As the reader might expect the answer heavily depends on the interpretation of probability. We show that if the fields are enumerated by the smallest height of their generators the probability is zero, at least if $d>6$. This is in contrast to what one expects when the fields are enumerated by the discriminant. The main result of this article is an estimate for the number of algebraic numbers of degree $d=e n$ and bounded height which generate a field that contains an unspecified subfield of degree $e$. If $n>\max\{e^2+e,10\}$ we get the correct asymptotics as the height tends to infinity.