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Counting number fields of fixed degree by their smallest defining polynomial

Santiago Arango-Piñeros, Fabian Gundlach, Robert J. Lemke Oliver, Kevin J. McGown, Will Sawin, Allechar Serrano López, Arul Shankar, Ila Varma

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 $\frac{27}{π^2} X^2$.

Counting number fields of fixed degree by their smallest defining polynomial

Abstract

When do two irreducible polynomials with integer coefficients define the same number field? One can define an action of on the space of polynomials of degree so that for any two polynomials and in the same orbit, the roots of may be expressed as rational linear transformations of the roots of ; thus, they generate the same field. In this article, we show that almost all polynomials of degree with size at most can only define the same number field as another polynomial of degree with size at most 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 fields such that the smallest polynomial defining the field has size at most is asymptotic to a constant times as long as . For , we obtain a precise asymptotic of the form .
Paper Structure (24 sections, 41 theorems, 140 equations, 2 figures)

This paper contains 24 sections, 41 theorems, 140 equations, 2 figures.

Table of Contents

  1. Introduction
  2. History and motivation
  3. Background
  4. Strategy of proof
  5. The quadratic case
  6. Setup
  7. Duke's theorem for negative discriminants
  8. Heights of imaginary quadratic fields
  9. The unit tangent bundle
  10. Duke's theorem for positive discriminants
  11. Heights of real quadratic fields
  12. Proof of Theorem quadratic case
  13. Special forms
  14. Forms with smaller Galois groups
  15. Forms whose discriminant is small
  16. ...and 9 more sections

Key Result

Theorem 1.1

The asymptotic size of the set of (isomorphism classes of) degree $n$ number fields $K/\mathbf{Q}$ with $\mathbf{P}^1$-height less than $X$ is for some constants $C_n>0$, as $X\to\infty$.

Figures (2)

  • Figure 1: Geometric interpretation of the geodesic and horocycle flows on $T_1\mathbf{H} =\mathrm{PSL}_2(\mathbf{R})$.
  • Figure 2: Geodesic corresponding to the quadratic form $x^2+xy-y^2$ of discriminant $5$.

Theorems & Definitions (86)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 2.1
  • Theorem 2.2: Duke
  • Corollary 2.3
  • Lemma 2.4
  • ...and 76 more