Abstract 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 Zx of degree n that are Eisenstein at the prime p. Order the polynomials by the natural height {Ht} (f). Define , ₍ (X) to be the proportion of polynomials f F, ₍ with {Ht} (f) X for which Heilbronn’s criterion applies. Then, align* ₗ , ₍ (X) \2{27, \;1- (p) \}, align* 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.
HIBBLER et al. (Mon,) studied this question.