Regular elliptic elements are dense in p-adic division algebra

Question

I'm trying to better understand the set $E$ of regular elliptic elements of $D^\times$, where $D$ is a finite dimensional central division algebra over a non-archimedean local field $F$.

For example, when $D$ has dimension $4$ over $F$, the set of regular elliptic elements is $D^\times \setminus F^\times$ but I don't have much of a feeling for what $E$ looks like beyond this case.

In particular, I would appreciate a reference, proof, or counterexample to the claim that:

$E$ is dense in $D^\times$.

This is alluded to in the proof of 1.4.5 of "The Essentially Tame Jacquet-Langlands Correspondence for Inner Forms of GL(n)" by Bushnell and Henniart, but I couldn't immediately see why this is true.

I'm interested in this because I would really like to understand the subset $\operatorname{Nrd}(E) \subset F^\times$, and ideally show that $$ \operatorname{Nrd}(E) = F^\times. $$

For example, this is true when $D$ has dimension $4$ over $F$. If someone could point me to a proof of this directly that would also be very much appreciated.

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Update: This follows directly from the claim that all elements of $D^\times$ are elliptic — in this case, the elliptic regular elements are the same as the regular elements, which are known to be dense (for example see the appendix to "Local tame lifting for GL(N). I : simple characters" by the same authors).

For example it is show in abx's answer to Regular or elliptic elements in the multiplicative group of central division algebra that all elements of $D^\times$ are elliptic when the characteristic of $F$ is $0$ (which is all I really care about) — so I just need to show their definition of regular and of elliptic regular agrees with that given in (1.4.5).

In (1.4.5) an element $h$ is called regular if the the reduced characteristic polynomial $f_h$ has no repeated roots over $\overline{F}$, and called elliptic regular if $h$ is regular and $f_h$ is irreducible (which as $F$ is separable is equivalent to requiring that $f_h$ is irreducible).

So is it possible to see directly that if $h$ is regular then $h$ is elliptic regular? I.e. that $f_h$ having no repeated roots over $\overline{F}$ implies that $f_h$ is irreducible?

Answer 1

If $m_h$ is the minimal polynomial of $h$ over $F$, then $f_h$ is a power of $m_h$. (This fact is standard, but, to have a definite reference, I looked though my books on central simple algebras and found it first as Lemma IV.2.4 of Berhuy and Oggier - An introduction to central simple algebras and their applications to wireless communication.) Therefore, since $f_h$ has no repeated roots, it equals $m_h$, hence is irreducible.