Counting number fields of fixed degree by their smallest defining polynomial

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 $\frac{27}{π^2} X^2$.

💡 Research Summary

The paper studies the problem of counting number fields of a fixed degree n by means of the smallest defining polynomial, measured by the maximum absolute value of its integer coefficients (the “P¹‑height”). The authors introduce a natural action of the group G = (ℚ^× × GL₂(ℚ))/T on the space Vₙ of binary n‑ic forms (i.e., homogeneous degree‑n polynomials in two variables with integer coefficients). Two forms f and g lie in the same G‑orbit precisely when the roots of f can be obtained from the roots of g by a rational linear change of variables; consequently, they generate the same number field K_f ≅ K_g.

The central technical result (Theorem 1.2) shows that, as the height bound X tends to infinity, the proportion of irreducible integer‑coefficient binary n‑ic forms of height ≤ X that share their field with another form of height ≤ X but are not G‑equivalent tends to zero. In other words, “almost all” such forms are uniquely determined (up to G‑equivalence) by the field they define. This strengthens earlier work of Bhargava, Shankar, and Wang, which only proved a positive‑proportion version.

To reach this conclusion the authors develop several new tools:

1. Binary‑form to ring correspondence. For each f∈Vₙ(ℤ) they construct a rank‑n ℤ‑algebra R_f generated by explicit linear combinations of powers of a root θ of f(x, 1). The discriminant of R_f equals the classical discriminant of f. This construction, originally due to Nakagawa and Wood, works for all binary n‑ic forms, not only monic ones.

2. Sₙ‑closure. Extending the work of Bhargava–Satriano, they define the Sₙ‑closure R_f^{(Sₙ)} of R_f, a rank‑n! ℤ‑module equipped with an Sₙ‑action. They give an explicit basis of R_f^{(Sₙ)} for arbitrary f, compute its discriminant, and show that for most f the basis vectors have comparable lengths. Consequently the lattice R_f^{(Sₙ)} is highly “skewed” (its successive minima differ by large factors), even though R_f itself is often not.

3. Successive minima and Minkowski theory. Using Minkowski’s second theorem together with the skewness of the Sₙ‑closure, they prove that for most f the filtration U_d(f) = ⟨1, θ, …, θ^d⟩_ℚ satisfies that the intersection R_f∩U_d(f) already provides a ℚ‑basis of U_d(f). In particular U₁(f) is the same for any two forms defining the same field, forcing the two defining polynomials to be related by the transformation f(x) ↦ a^{−n} f(ax + b) (a∈ℚ^×, b∈ℚ), i.e., they lie in the same G‑orbit.

4. Counting arguments. They combine the above structural results with classical density theorems: Hilbert’s irreducibility theorem guarantees that a density‑one set of forms has full symmetric group Sₙ as Galois group; the discriminant being a homogeneous polynomial of degree 2(n−1) yields Disc(f)≈X^{2(n−1)} for height ≤ X; and recent work (BSW25) shows that for most f the discriminant is essentially square‑free, so R_f is of index 1 or 2 in the full ring of integers of K_f. A sieve‑theoretic analysis then shows that the number of G‑orbits of irreducible forms of height ≤ X is asymptotic to a positive constant Cₙ X^{n+1}.

Putting everything together, the authors obtain the main counting theorem (Theorem 1.1):

• For n = 2, the number N₂(Ht < X) of quadratic fields whose minimal defining polynomial has height ≤ X is exactly \