Polynomial densities and Heilbronn's criterion

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 (p-1,n)=1. Let Fp,n denote the collection of monic polynomials f[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 fp,n with Ht(f)≤ X for which Heilbronn's criterion applies. One has X→∞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 (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.