Totally p-adic Numbers of Degree 3

TL;DR

This paper investigates the minimal height of degree 3 totally p-adic algebraic numbers and provides an algorithm to compute this minimal height for any prime p.

Contribution

It introduces a novel algorithm to determine the smallest height of degree 3 totally p-adic numbers for a given prime p.

Findings

Algorithm successfully computes minimal heights for various primes p.

Provides insights into the distribution of heights among degree 3 totally p-adic numbers.

Establishes bounds and properties related to the height of these algebraic numbers.

Abstract

The height of an algebraic number $\alpha$ is a measure of how arithmetically complicated $\alpha$ is. We say $\alpha$ is totally $p$-adic if the minimal polynomial of $\alpha$ splits completely over the field $\mathbb{Q}_p$ of $p$-adic numbers. In this paper, we investigate what can be said about the smallest nonzero height of a degree $3$ totally $p$-adic number. In particular, we provide an algorithm to determine, given a prime $p$, the smallest height of a degree $3$ totally $p$-adic number.