#### Vol. 4, No. 1, 2020

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Totally $p$-adic numbers of degree 3

### Emerald Stacy

Vol. 4 (2020), No. 1, 387–401
##### 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 ${ℚ}_{p}$ of $p$-adic numbers. We investigate what can be said about the smallest nonzero height of a degree $3$ totally $p$-adic number.

##### Keywords
height, algorithm, $p$-adic
##### Mathematical Subject Classification 2010
Primary: 11G50, 11S20, 11Y40, 12Y05
##### Supplementary material

Abelian cubic polynomials and congruence classes (mod $m_i$) for splitting over $mathbb{Q}_p$