POLYNOMIAL DENSITIES AND HEILBRONN’S CRITERION

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 $K=\mathbb {Q}[x]/(f)$ generated by f . Suppose $n\geq 3$ is odd and $p\geq 5$ is prime with $\gcd (p-1,n)=1$ . Let $\mathcal {F}_{p,n}$ denote the collection of monic polynomials $f\in \mathbb {Z}[x]$ of degree n that are Eisenstein at the prime p . Order the polynomials by the natural height $\operatorname {\mathrm {Ht}}(f)$ . Define $\delta _{p,n}(X)$ to be the proportion of polynomials $f\in \mathcal {F}_{p,n}$ with $\operatorname {\mathrm {Ht}}(f)\leq X$ for which Heilbronn’s criterion applies. Then, $$ \begin{align*} \liminf_{X\to\infty} \delta_{p,n}(X)\geq \max\{\tfrac{2}{27},\;1-\varepsilon(p)\}, \end{align*} $$ where $\varepsilon (p)\to 0$ and is effectively computable. In particular, the lower density tends to $1$ as $p\to \infty $ 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 $n\geq 2$ , a positive proportion of Eisenstein polynomials of degree n fail to generate norm-Euclidean fields.