Counting number fields of fixed degree by their smallest defining polynomial

TL;DR

This paper proves that most degree n polynomials with bounded size define distinct number fields unless they are related by a specific group action, leading to an asymptotic count of such fields as size grows.

Contribution

It establishes that almost all polynomials of degree n with bounded size generate unique number fields unless they are in the same orbit under a specific group action, refining previous results.

Findings

Number of degree n fields with smallest defining polynomial size ≤ X is asymptotic to a constant times X^{n+1} for n ≥ 3.

For quadratic fields (n=2), the count is asymptotic to (27/π^2) X^2.

Almost all polynomials of bounded size define distinct fields unless related by the group action.

Abstract

When do two irreducible polynomials with integer coefficients define the same number field? One can define an action of $\mathrm{GL}_2 \times \mathrm{GL}_1$ on the space of polynomials of degree $n$ so that for any two polynomials $f$ and $g$ in the same orbit, the roots of $f$ may be expressed as rational linear transformations of the roots of $g$; thus, they generate the same field. In this article, we show that almost all polynomials of degree $n$ with size at most $X$ can only define the same number field as another polynomial of degree $n$ with size at most $X$ if they lie in the same orbit for this group action. (Here we measure the size of polynomials by the greatest absolute value of their coefficients.) This improves on work of Bhargava, Shankar, and Wang, who proved a similar statement for a positive proportion of polynomials. Using this result, we prove that the number of degree $n$ fields such that the smallest polynomial defining the field has size at most $X$ is asymptotic to a constant times $X^{n+1}$ as long as $n\geq 3$. For $n = 2$, we obtain a precise asymptotic of the form $\frac{27}{\pi^2} X^2$.