# Linear relations with conjugates of a Salem number

## TL;DR

This paper investigates linear relations among conjugates of Salem numbers, revealing their connection to totally real algebraic integers and establishing minimal degree and length constraints for such relations.

## Contribution

It demonstrates that all linear relations among Salem conjugates derive from relations among conjugates of a related totally real algebraic integer and determines minimal degree and length bounds.

## Key findings

- All relations derive from conjugates of α+1/α
- Minimal degree of Salem number with nontrivial relation is 8
- Minimal length of a nontrivial relation is 6

## Abstract

In this paper we consider linear relations with conjugates of a Salem number $\alpha$. We show that every such a relation arises from a linear relation between conjugates of the corresponding totally real algebraic integer $\alpha+1/\alpha$. It is also shown that the smallest degree of a Salem number with a nontrivial relation between its conjugates is $8$, whereas the smallest length of a nontrivial linear relation between the conjugates of a Salem number is $6$.