2025/12/18 by Hibbler, Alexis, Kevin J. McGown, McGown, Kevin J. +2
Mathematics · #11A05 #11C08 #11D07 #11R04 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2512.16220
openalex publication_date 2025/12/18 · openalex created_date 2025/12/21 · openalex updated_date 2026/07/31
Heilbronn gave a sufficient condition for a number field with a totally ramified prime to fail to be norm-Euclidean. We say that Heilbronn's criterion applies to a polynomial f if it applies to the number field K=ℚ[x]/(f) generated by f. Suppose n≥ 3 is odd and p≥ 5 is prime with gcd(p-1,n)=1. Let Fp,n denote the collection of monic polynomials f∈ℤ[x] of degree n that are Eisenstein at the prime p. We order our polynomials by the natural height Ht(f). Define δp,n(X) to be the proportion of polynomials f∈ Fp,n with Ht(f)≤ X for which Heilbronn's criterion applies. One has \liminfX→∞δp,n(X)≥ max\(2)/(27) , 1-ε(p)\ , where ε(p)→ 0 and is effectively computable. In particular, the lower density tends to 1 as p→∞ uniformly in n. We also give a version of this result where we weaken the condition on gcd(p-1,n). As a corollary, we show that given an integer n≥ 2, a positive proportion of Eisenstein polynomials of degree n fail to generate norm-Euclidean fields.