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=Qx/ (f) generated by f. Suppose n 3 is odd and p 5 is prime with (p-1, n) =1. Let F₏, ₍ denote the collection of monic polynomials f of degree n that are Eisenstein at the prime p. We order our polynomials by the natural height Ht (f). Define δ₏, ₍ (X) to be the proportion of polynomials f₏, ₍ with Ht (f) X for which Heilbronn's criterion applies. One has ₗδ₏, ₍ (X) \2{27\, , \;1- (p) \}\, , where (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.